Combo with "8 Pythagorean Theorem" and 27 others
Reciprocal
2 numbers are reciprocals if their product is 1.
Surface area of a right circular cylinder
2(pi*(r^2))+ 2*pi*r*h
20<all primes<30
23, 29
Translate the sentence in to an equation. Twice a increased by the cube of a equals b
2a + a³ = b
2⁵/2³
2²
Convert 60% to a fraction
3/5
30< all primes<40
31, 37
Write the equation in standard form: y − 11 = 3(x − 2)
3x − y = −5
If 4500 is invested at a simple interest rate of 6%, what is the value of the investment after 10 months?
4725
Find two consecutive even integers whose sum is 126.
62, 64
5/8 in percent?
62.5%
Evaluate the expression 5 raised to the fourth
625
What are the smallest three prime numbers greater than 65?
67, 71, 73
Find the degree of the polynomial 5x²y + 7xy⁶− 3xy
7
heptagon
7-sided polygon
Simplify the expression. If not possible, write simplified 3x + 6x
9x
normal distribution
A bell shaped probability distribution. There are as many values less than the mean as there are values greater than the mean.
compound events
A combination of simple events.
number line
A diagram that represents numbers as points on a line.
Quadrilateral
A four sided polygon
Equation
A mathematical sentence that contains an equal sign
variance
A measure of dispersion of data centered about the mean.
∅ is
A multiple of every integer
Proportion
A proportion is an equation relating two ratios;
quantitative change
A relationship that can be expressed in numerical terms.
System of inequalities
A set of two or more inequalities with the same variables
protractor
A tool used to measure angles
Right triangle
A triangle with one right triangle
coordinate grid (Cartesian)
A two-dimensional system in which a location is described by its distances from two intersecting, usually perpendicular, straight lines called axes.
Area of a triangle
A= (1/2) b*h
Area of a triangle
A= (1/2)b*h
parallel
Always the same distance apart.
quantity
An amount.
Right Angle
An angle that measures 90 degrees.
Right angle
An angle that measures 90°
Acute Angle
An angle that measures between 0 and 90 degrees.
conjecture
An educated guess or opinion
proportion
An equation showing that two ratios are equal.
Boundary
An equation that divides the coordinate plane into two half planes
Counterexample
An example that proves that a conjecture or statement is false.
What is the name of set with a number of elements which cannot be counted?
An infinite set.
multiples of 3
An integer is divisible by 3 if the sum of its digits is divisible by 3
Multiples of 4
An integer is divisible by 4 if the last two digits form a multiple of 4
multiples of 9
An integer is divisible by 9 if the sum of its digits is divisible by 9
factor
An integer that divides evenly into another.
Included angle
Angle formed by two adjacent sides of a polygon.
Alternate exterior angles
Angles on opposite sides of the transversal t, outside lines r and s
Base
Any side of a triangle
Formula to calculate arc length?
Arc length = (n/360) x pi(2r) where n is the number of degrees.
Area of a Sector
Area of a Sector = (central angle/360) x πr²
A=πr²
Area of a circle
bh
Area of a parallelogram
½d₁d₂
Area of a rhombus
THE DENOMINATOR CAN NEVER
BE ZERO! 1/∅=null
Formula to find a circle's circumference from its diameter?
C = (pi)d
Circle Graphs
Circle graphs, often called pie charts, are used to represent data with a relatively small number of categories. They illustrate how a whole is separated into parts. The area of the circle graph representing each category is proportional to the part of the whole that the category represents.
composite functions
Combination of two functions, where you apply the first function and get an answer, and then fill that answer into the second function.
nCr+(n!)÷((n-r)×r!)
Combinations
Line Bisector
Cuts a line segment into two congruent segments.
Angle Bisector
Cuts an angle into two congruent angles.
Cylinder
Cylinder a three-dimensional figure with two parallel, congruent circular basses connected by a curved lateral surface.
State whether the percent of change is a percent of increase or percent of decrease. Find the percent of change to the nearest whole percent. Original $25, New $10
Decrease of 60%
relative position
Determines location of a number when comparing numbers (5 is between 1 and 10 or 6 is less than 8).
Circumference of a circle?
Diameter(Pi)
Arithmetic sequence
Difference between successive terms is constant
Vertical lines
Do not have slopes!
Expression in simplest terms
Does not contain any like terms or parenthesis
∅ Is
EVEN
Pressure (Nm-2 / Pa)
F / A
similar (figures)
Figures that have the same shape, but not necessarily the same size.
common denominator
For two or more fractions, a common denominator is a common multiple of the denominators.
Between
Given 3 points, A, B, and C, B is between A and C iff all three of the points lie on the same line
converse
If q, then p The antecedent and the consequent switch places
Triangle rigidity
If the side lengths of a triangle are given, the triangle can have only one shape
identity property of addition
If you add zero to a number the sum is the same as that number.
Base
In an expression of the form xⁿ, the x is called the _________
inference
Judge whether the number you found is the number you expected.
Vertex of a parabola
Lowest point on the graph of a quadratic function that opens upward or highest point of a quadratic graph that opends downward
EH
M x c x (T1 - T2)
Line
Made up of an infinite number of points. Has infinite length, but no thickness or width.
Best fit line
Many calculators use an algorithm called linear regression to find a more precise line of fit
amplitude
Maximum displacement of wave
Numerical Expression
May contain only constants and operations.
Algebraic Expression
May contain variables, constants, and operations.
Kilo, Hecto, Deka, Base, Deci, Centi, Milli
Metric
a/∅
Null
x-axis
On a coordinate grid, the horizontal axis.
Total Pressure
P of liquid + P of atmosphere
nPr=(n!)÷((n-r)!)
Permutations
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 3x + 5y = 10 and 5x − 3y = −6
Perpendicular
a²+b²=c²
Pythagorean Theorum
Graph y > −2x + 1 and y ≤ x + 3
See how the graph should appear on page 382
y=mx+b
Slope intercept form
Coordinate proof
Style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane.
Distance
The absolute value of the difference of the coordinates of the point
Volume
The amount of space enclosed by a solid (three-dimensional) figure.
What is the "domain" of a function?
The set of input values for a function.
What is the "range" of a function?
The set of output values for a function.
Inverse
The statement formed by negating the hypothesis and conclusion of a conditional statement
Concurrent
Three or more lines that intersect at one point
Period
Time taken for one complete vibration (oscillation).
Compound inequality
Two inequalities put together with the word and or two inequalities put together with the word or
V
U +- at
V2
U2 +- 2as
Find the slope of the line that passes through (5, 2) and (5, −2)
Undefined
Linear extrapolation
Use of a linear equation to make predictions about values that are beyond the range of the data
Gay Lussac's Pressure Law
Volume is constant
0, 1, 2, 3, 4 ...
Whole numbers
integers
Whole numbers and their opposites (. . . -3, -2, -1, 0, 1, 2, 3. . .)
Integers
Whole numbers and their opposites.
Standard form of a polynomial
Written in order from greatest degree to least degree
5x^2 - 35x -55 = 0
[(7+ sqrt93) /2], [(7 - sqrt93) / 2]
Solve 2(a − 3) = 3(−2a + 6)
a = 3
Solve the equation for the variable a listed . . . 7a − b = 15a for a
a = −b/8
equation
a mathematical sentence that describes a relationship between two expressions that are equal.
variable
a symbol, usually a letter, used to represent an unknown or changing quantity in mathematical expressions or equations.
c) dimensional sizing
a) An instrument used for weighing.
f(x) = x² + 3 Find f(b) + 4
b² + 7
bⁿ
b∧b∧b (where b is used as a factor n times)
Circumference of a Circle
c=2 x pi x r OR pi x D
Circumference of a Circle
c=2*pi*r OR d*pi
What is the common monomial factor in the expression 4(c^3)d - (c^2)(d^2) + 2cd?
cd
conditional statement
compound statement written in the "if-then" form.
E(h)
cΔT
Rate
d/t (distance)/(time)
Find distance when given time and rate
d=rt so r= d/t and t=d/r
consecutive (adjacent) vertices
endpoints of a side
A polygon with ll sides congruent
equilateral
Horizontal
from left to right or right to left.
X-axis
horizontal line in a coordinate plane
A number is divisible by 6 if...
its divisible by 2 and by 3.
A number is divisible by 4 is...
its last two digits are divisible by 4.
If the two sides of a triangle are unequal then the longer side.................
lies opposite the greater angle
EH
m x l x f
plot
mark points on a graph
Simplify m⁴r²/mr⁴
m³/r²
A polygon with 8 sides
octagon
A polygon with 5 sides
pentagon
magnitude
size and scale reflected by a value.
1 is the
smallest positive integer
sum of an arithmetic sequence of N consecutive numbers
sum = N x (first number + last number) / 2
average/mean
sum of all the values in a data set divided by the number of data values
like terms
terms in an expression that have the EXACT SAME variables for example in the expression: 4x - 2y + 3x - 7xx 4x and 3x are the only like terms
The larger the absolute value of the slope...
the steeper the slope.
A polygon with three sides
triangle
twice
two times as much
S
ut +- 1/2a (t2)
Mechanical Efficient
work output / work input
Solve the equation 4x − 1 = 0
x = 1/4
Slope
y₂-y₁/x₂-x₁
Any Horizontal line slope
zero
Solve and graph |x − 4|< 4
{x | 0 < x < 8} The number line has an open circle on zero and an open circle on eight with shading between
Solve −8x < −64
{x | x > 8}
n
λ1 / λ2 OR real depth / apparent depth
If a product of two numbers is ∅, one number must be
∅
∅ divided by 7
∅
Find the slope of the line that passes through (10, 0) and (−2, 4)
−1/3
f(x) = 2x − 4 Find the value of f(−5)
−14
Solve 2/3 n + 8 = 1/3 n + 2
−18
Simplify (−3ab⁴)³
−27a³b¹²
Backpack $56.25. Discount 20%. Find the discounted price
$45.00
Dress $69 and tax is 5 %. Find the total price
$72.45
R
( 1/R1) + (1/R2)
What point does every direct variation equation go through?
(0, 0)
Use substitution to solve the system of equations: y = 4x and x + y = 5
(1, 4)
Use elimination to solve the system of equations: x + 4y = 11 and x − 6y = 11
(11, 0)
Use an augmented matrix to solve the system of equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2x − y = 7 and −x + 3y = −11
(2, −3)
Factor 4x²− 25
(2x − 5)(2x + 5)
2³×7³
(2x7)³
Factor a^2 + 2ab + b^2
(a + b)^2
a^2 + 2ab + b^2
(a + b)^2
How to determine percent decrease?
(amount of decrease/original price) x 100%
Area of a triangle?
(base*height) / 2
diagonal
(geometry) a straight line connecting any two vertices of a polygon that are not adjacent
Volume of a rectangular solid
(length)(width)(height)
Factor n² + 7n + 12
(n + 4)(n + 3)
Area of a circle
(pi)r²
Distance
(rate)(time) d=rt
Factor 2t² + 9t − 5
(t+ 5)(2t −1)
Fator 2x² + 5x + 2
(x + 2)(2x + 1)
binomial product of (x+y)²
(x+y)(x+y)
binomial product of (x-y)²
(x+y)(x-y)
Use an augmented matrix to solve the system of equations: x + 4y = 19 and −3x − 2y= −7
(−1 , 5)
Solve using elimination: . . . . −3x −4y = −1 and 3x− y =−4
(−1, 1)
Solve the system of equations by graphing . . . . x + 3y = −3 and x − 3y = −3
(−3, 0)
Solve using elimination: . . . . . . . . . . . . . . . . −3x −4y = −4 and x + 3y = −1
(−4, −2)
Kelvin
+273 Degrees Celcius
If a>b then
-a<-b
A parallelogram
...
a) angle measurement
...
b) A measurement of hotness or coldness (Celsius, Fahrenheit, Kelvin)
...
b) The possible values for y in function.
...
degrees
...
Find the slope of the line that passes through (6, 1) and (−6, 1)
0
What is the slope of a horizontal line?
0
Solve (4x + 7)/15 = (6x +2)/10
0.8
1ⁿ
1
The product of any number x and its reciprocal
1
a^0 =
1
b¹
1
Evaluate 4/11 + 11/12
1 & 37/132
Characteristics of a Rectangle
1) Opposite sides are equal 2) Diagonals are equal
Perfect Squares 1-15
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Side lengths of a 45-45-90 right triangle
1-1-sqrroot of 2
Side lengths of a 30-60-90 right triangle
1-sqrroot of 3-2
Write 10,843 X 10^7 in scientific notation
1.0843 X 10^11
8.84 / 5.2
1.7
Look at problem 17 on page 176. You will see a graph. What is the slope of the graph?
1/2
Kinetic energy (Ek)
1/2 (mv2)
1/2 divided by 3/7 is the same as
1/2 times 7/3
S
1/2(V+U) t
Convert 33.33% to a fraction
1/3
The reciprocal of any non-zero #x is
1/x
The reciprocal of any non-zero number is
1/x
The negative exponent x⁻ⁿ is equivalent to what?
1/xⁿ i.e. 5^-3 = 1/(5^3) = 1/ 125 = .008
Legs 6, 8. Hypotenuse?
10
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. In how many ways can the judges award the 3 prizes?
10! / (10-3)! = 720
decagon
10-sided polygon
Write an algebraic expression for the product of ten and x
10x
Find three consecutive integers whose sum is 36
11, 12, 13
Evaluate if x = −1, y = 3, and z = −4. . . . . . . . |−3y + z| − x
14
Find the next three terms of the arithmetic sequence 22, 20, 18, 16 ...
14, 12, 10
Simplify the expression. If not possible write simplified 7(2x + 5)
14x + 35
Find the degree of the polynomial 2x² + 3x + 7
2
Define slope
(y2 − y1)/(x2 − x1) Other definitions are rise/run and change in y/change in x
Celcius
-273 K
3 is the opposite of
-3
A cylinder has surface area 22pi. If the cylinder has a height of 10, what is its radius?
1
y = 2x The slope is ____________
2
Simplify 9^(1/2) X 4^3 X 2^(-6)?
3
If a lamp decreases to $80, from $100, what is the decrease in price?
= (actual decrease/Original amount) x100% = 20/100x100% = 20%
percentile
A division of ordered data into 100 equal parts. About 1% of the data are in each part.
sketch
A drawing completed quickly, but still recognizable.
Flowchart proof
A style of proof that uses boxes and arrows to show the structure of the proof.
What is the name for a grouping of the members within a set based on a shared characteristic?
A subset.
What is a tangent?
A tangent is a line that only touches one point on the circumference of a circle.
Define a "term",
A term is a numerical constant or the product (or quotient) of a numerical constant and one or more variables. (3x, 4x^2 and 2a/c)
Undefined term
A term that cannot be defined by using other figures, a building block of geometry
Order of operations
Evaluate expressions inside grouping symbols Evaluate all powers Multiply and/or divide from left to right Add and/or subtract from left to right
Is 0 even or odd?
Even
∅ is
Even
2 is the only
Even prime number
Quadrilaterals
Every quadrilateral has four sides and four interior angles. The measures of the interior angles add up to 360o.
P(event)=(Number of successes)÷(Number of trials)
Experimental Probability
Is the equation y = 5ⁿ linear, quadratic, or exponential?
Exponential
Exponents
Exponents are used to denote the repeated multiplication of a number by itself;
Scientific notation
Expressed as a X 10ⁿ , where a is the integer 1, 2, 3, 4, 5, 6, 7, 8, or 9 and n is an integer
Factored form of a monomial
Expressed as the product of prime numbers and variables, and no variable has an exponents greater than one
Median of a triangle
Segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Radius
Segment whose endpoints are the center of the circle and a point on the circle.
Geometric sequence
Sequence in which each term after the first is found by multiplying the previous term by a nonzero constant called the common ratio
Step function
Series of line functions also called a piecewise-linear function
Ordered pair
Set of numbers or coordinates written in the form (x, y)
Locus
Set of points that satisfies a given condition
Range
Set of second numbers of the ordered pairs. The y values
Intersection
Sets can be formed from other sets. If S and T are sets, then the intersection of S and T is the set of all elements that are in both S and T and is denoted by S∩T.
Parabola
Shape of the graph of a quadratic function
What is the "solution" for a set of inequalities.
The overlapping sections.
Sketch a quadratic equation that has no roots (solutions)
The parabola does not touch the x-axis. See page 537.
Sketch a quadratic equation that has one root (solution).
The parabola touches the x-axis at one point. See page 537
Hypothesis
The part of a conditional statement following the word if.
Conclusion
The part of a conditional statement following the word then
Segment
The part of a line consisting of two points and all points between them
Perimeter
The perimeter of a polygon is the sum of the lengths of its sides. The area of a polygon refers to the area of the region enclosed by the polygon.
Base (of an isosceles triangle)
The side opposite of the vertex of an isosceles triangle.
What is true about the slopes of parallel lines?
The slopes of parallel lines are the same
Find (n −4)(n − 6)
n² − 10n + 24
Solve 5(x + 3)+9= 3(x − 2) + 6
x = −12
Given the point (2, 2) and m = −3, write the equation in point slope form
y − 2 = −3(x − 2)
Look at the graph for problem #40 on page 235. Write the equation in point slope form
y − 3 = 4(x − 1)
Given the point (−8, 5) and m = −2/5, write the equation in point slope form
y − 5 = −2/5(x + 8)
Point-slope form
y − y₁ = m (x − x₁)
#2 What are the important properties of a 45-45-90 triangle?
• The triangle is isosceles (AC=BC).
200 <_ x <_ 300. How many values of x are divisible by 5 & 8?
3
Evaluate 162 ÷ [6(7 − 4)²]
3
In the equation y = 3x + 7, the slope is _______________
3
What is the measure of an exterior angle of a regular pentagon?
72
What number between 70 & 75, inclusive, has the greatest number of factors?
72
Find the product (3x − 2)(3x + 2)
9x² − 12x + 4
²Simplify. Assume no denominator is equal to zero. (4x/3x²)−²
9x²/16
Find the product (3x − 1)²
9x²− 6x + 1
ray
A part of a line that has one endpoint and goes on forever in one direction.
Ray
A part of a line that starts at an endpoint and extends forever in one direction
line segment
A part of a line with two endpoints.
What is a percent?
A percent is a fraction whose denominator is 100.
What is a minor arc?
The shortest arc between points A and B on a circle's diameter.
Maximum
Vertex of a quadratic graph that opens downward
Minimum
Vertex of a quadratic graph that opens upward
Velocity Ratio
distance moved by object/distance moved by load
formula for distance problems
distance=rate×time or d=rt
Frequency
number of complete vibrations per second.
area
number of square units needed to fill up a region on a
fraction
numerator divided by the denominator
(12sqrt15) / (2sqrt5) =
(12/2) x (sqrt15 / sqrt5) = 6sqrt3
Square of a sum
(a + b)² = a² + 2ab + b²
a^2 - b^2
(a - b)(a + b)
Express the number 158 X 10⁻⁷in scientific notation
1.58 X 10⁻⁵
Express 1,900,000 in scientific notation
1.9 X 10⁶
Convert 25% to a fraction
1/4
Convert 20% to a fraction
1/5
Convert 16.66% to a fraction
1/6
Convert 12.5% to a fraction
1/8
(a^-1)/a^5
1/a^6
(2²)³
2⁶
Choc chip cookies sell for $6.95 and white choc cookies sell for $5.95 per lb. How many pounds of choc chip cookies must be mixed with 4 lbs of white choc cookies to obtain a mixture that sells for $6.75 per pound
16 pounds
Factor completely 3a² + 30a + 63
3(a + 7)(a + 3)
Quadrilateral
4-sided polygon
Convert 80% to a fraction
4/5
Employee X is paid 19.50 per hour no matter how many a week. Employee Y earns 18 for the first 40 and 1.5 the hourly wage for every hour after that. If both earned the same amount and worked the same in one week, how many did each work?
48
Evaluate 6²+ 3 ∙ 7 − 9
48
If Madagascar's exports totaled 1.3 billion in 2009, and 4% came from China, what was the value in millions of the country's exports to China?
52
Evaluate if x = 2, y = 3, z = 4 2xyz + 5
53
50 < all primes< 60
53, 59
Determine the next three terms in the geometric sequence 2, 6, 18 . . .
54, 162, 486
What is the area of a regular hexagon with side 6?
54sqrt3. (divide the hexagon into 6 congruent equilateral triangles.
60 < all primes <70
61, 67
Straight angle
A 180° angle
1:sqrt3:2 is the ratio of the sides of what kind of triangle?
A 30-60-90 triangle.
Area of a rectangle
A = length x width
Formula for the area of a circle?
A = pi(r^2)
histogram
A bar graph in which the labels for the bars are consecutive groups of numbers.
What is a central angle?
A central angle is an angle formed by 2 radii.
Transformation
A change in the position, size, or shape of a figure or graph
ratio
A comparison of two numbers or measures using division.
solid figure
A figure with three dimensions.
Point of concurrency
A point where three or more lines coincide
exponential notation
A way of writing numbers using exponents.
Area of a trapezoid
A= (1/2)h*(a+b) where a is the length of the bottom base and b is the length of the top base.
Area of parallelogram
A=b*h
Area of a rectangle
A=l*w
Area of a circle
A=pi*(r^2)
table
An arrangement of information or data into columns and rows or a condensed list.
finding the average of evenly spaced numbers
Average the smallest and the largest
CPCTC
Abbreviation for the phrase "Corresponding parts of congruent triangles are congruent."
When multiplying exponential #s with the same base, you do this to the exponents...
Add them. i.e. (5^7) * (5^3) = 5^10
When multiplying exponential #s with the same base, you do this to the exponents...
Add them. i.e. (5^7) * (5^3) = 5^10
Standard form
Ax + By = C
correlation
An association between two variables used in statistics.
Define a "monomial"
An expression with just one term (-6x, 2a^2)
Power
An expression written with an exponent and a base.
Exterior Angles of a Triangle
An exterior angle of a triangle is equal to the sum of the remote interior angles. The three exterior angles of a triangle add up to 360 degrees.
Remote interior angle
An interior angle that is not adjacent to the exterior angle.
1:1:sqrt2 is the ratio of the sides of what kind of triangle?
An isosceles right triangle.
Measure
Angles are measured in degrees
Corresponding angles (of polygons)
Angles in the same position in polygons with an equal number of sides.
congruent angles
Angles that have the same measure
common fraction
Any fraction whose numerator and denominator is a common multiple of the denominators.
Zero exponent property
Any nonzero number raised to the zero power is equal to 1
½(b₁+b₂)h
Area of a trapezoid
A=½bh
Area of a triangle
4πpw
Area of circular ring
Determine whether the sequence is arithmetic, geometric, or neither 1, −5 −11, −17 . . .
Arithmetic
Three-Dimensional Figures
Basic three-dimensional figures include rectangular solids, cubes, cylinders, spheres, pyramids, and cones.
Binary multiplication
Binary multiplication is as simple as multiplication in decimal system.
Formula to find a circle's circumference from its radius?
C = 2(pi)r
For any number x
Can be negative, zero, or positive
Prime polynomial
Can not be written as a product of two polynomials with integral coefficients
quantitative
Capable of being measured or expressed in numerical terms.
A=(1+r/n)ⁿt
Compound interest
Algebraic expression
Consists of sume and/or products of numbers and variables
Ordered Pair
Consists of two numbers. One is on the x-axis, and the other on the y-axis.
Look at problem 11 on page 183. Name the constant of variation. What is the slope?
Constant of variation is −5 The slope is −5
How do you solve proportions? a/b=c/d
Cross multiplication a/b=c/d 4/6=10/15 4(15)=6(10) 60=60
Line of fit
Data points all lie close to a line that models the trend
Bivariate data
Data with two variables
A polygon with 10 sides
Decagon
Is the equation y = 300(.95)t exponential growth or decay
Decay
n!
Factorial
Congruent line segments
Given any two points on a line, a line segment is the part of the line that contains the two points and all the points between them. The two points are called endpoints. Line segments that have equal lengths are called congruent line segments.
Combinations
Given the five letters A, B, C, D, and E, suppose that you want to determine the number of ways in which you can select 3 of the 5 letters, but unlike before, you do not want to count different orders for the 3 letters. The following is a list of all of the ways in which 3 of the 5 letters can be selected without regard to the order of the letters. ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE
Solve the equation 15x −30 = 5x − 50 by graphing and algebraically
Graph f(x) = 10x + 20 and it crosses the x-axis at − 2. Solve 10x + 20 and x = −2
Discrete function
Graph that consists of values that are not connected
Continuous function
Graphed with a line or smooth curve
Degree of a polynomial
Greatest degree of any term in a polynomial
Sketch a system of equations that has one solution.
See the concept summary on page 333.
Identify the hypothesis and conclusion for the following relation: If it is Sunday, then mail is not delivered
Hypothesis: It is Sunday Conclusion: mail is not delivered
P
I x V
Q
I x t
P
I2 x V
Parallel lines
Lines in the same plane that do not intersect.
Skew lines
Lines that are not coplanar
Perpendicular lines
Lines that intersect at a right angle
If Event is impossible
P(E) = ø
Pressure1/t1
P2/t2
Pressure1Volume1
P2V2
Perimeter of a rectangle
P= 2L + 2w
The Perimeter of a rectangle
P=2(l+w)
The Perimeter of a Square
P=4s (s=side)
What is the order of operations?
PEMDAS (Parentheses Exponents Multiplication/Division Addition/Subtraction)
Coordinate plane
Plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) The location, or coordinates, of a point are given by the ordered pair (x, y)
P=I-E
Profit
directly proportional quantities
Quantities in which ratios are constant; in other words, if one increases, the ones "directly proportional" with it also will increase. This also can be defined as quantities related such that if one increases, the other also will increase proportionally.
Percent of change
Ratio of change in an amount to the original amount expressed as a percent
Slope
Ratio of the change in the y-coordinates (rise), to the change in the x-coordinates (run) as you move from one point to another
Rate
Ratio of two measurements having different units of measure
Describe how the graph of the function is related to the graph of f(x) = x² g(x) = −2x²
Reflected over the x-axis and stretched vertically
Regular pyramid
Regular pyramid a pyramid whose base is a regular polygon and whose lateral faces are all congruent.
Inverse of a relation
Relation obtained by switching the coordinates in each ordered pair
Function
Relationship between input and output where there is exactly one output for each input
Which is greater? 200x^295 or 10x^294?
Relationship cannot be determined (what if x is negative?)
Distributions of Data
Relative frequency distributions given in a table or histogram are a common way to show how numerical data are distributed. In a histogram, the areas of the bars indicate where the data are concentrated.
x-coordinate
Represents the horizontal placement of the point
y-coordinate
Represents the vertical placement of the point
Multi-step equation
Required more than one step to solve
Simple interest
Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed called the principal, for the entire time period. If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula V=P(1+rt/100)
In the equation, y = mx + b, the m stands for _____________
Slope
m=(y₂-y₁)÷(x₂-x₁)
Slope
πr²+rπl
Surface area of a cone
6b
Surface area of a cube
2πr²+2πrh
Surface area of a cylinder
2b+ph
Surface area of a prism
b+½pl
Surface area of a pyramid
4πr²
Surface area of a sphere
Surface area
Surface area the sum of the areas of the faces, or surfaces, of a three-dimensional figure.
relational symbols
Symbols included are <, >, ≤, ≥, ≠, =.
operational symbols
Symbols representing the operations of addition, subtraction, multiplication and division.
Variables
Symbols to represent unspecified numbers or values
natural numbers
The counting numbers; 1, 2, 3, 4...
X-coordinate
The first number in an ordered pair; it lies on the horizontal line.
Distance formula
The formula: d =√(x₂ − x₁)² + (y₂ − y₁)²
Midpoint formula
The formula: m = (x₁ + x₂)/2, (y₁ + y₂)/2
quadrant
The four sections of a coordinate grid that are separated by the axes.
Describe the relationship between 3x^2 and 3(x - 1)^2
The graph of 3(x - 1)^2 is a translation (shift) of the graph one unit or space to the right.
Completing the square
To add a constant term to a binomial of the form x² + bx so that the resulting trinomial is a perfect square. One method to solve quadratic equations
Bisect
To divide into two congruent parts
Quotient of powers
To divide two powers with the same base, subtract the exponents
compute
To find a numerical result, usually by adding, subtracting, multiplying or dividing.
Power of a power
To find the power of a power, multiply the exponents
Power of a product
To find the power of a product, find the power of each factor and multiply
Power of a quotient
To find the power of a quotient, find the power of the numerator and the power of the denominator
Elimination
Use addition or subtraction to eliminate one variable and solve a system of equations
Substitution
Use algebraic methods to find an exact solution of a system of equations
one to one correspondence
Used to compare two sets in which one element matches one and only on element in the other set.
Histograms
When a list of data is large and contains many different values of a numerical variable, it is useful to organize it by grouping the values into intervals, often called classes. To do this, divide the entire interval of values into smaller intervals of equal length and then count the values that fall into each interval. In this way, each interval has a frequency and a relative frequency. The intervals and their frequencies (or relative frequencies) are often displayed in a histogram. Histograms are graphs of frequency distributions that are similar to bar graphs, but they have a number line for the horizontal axis. Also, in a histogram, there are no regular spaces between the bars. Any spaces between bars in a histogram indicate that there are no data in the intervals represented by the spaces.
Percent Change
When a quantity changes from an initial positive amount to another positive amount. This is called percent change.
Random Variables
When analyzing data, it is common to choose a value of the data at random and consider that choice as a random experiment. Then, the probabilities of events involving the randomly chosen value may be determined. Given a distribution of data, a variable, say X, may be used to represent a randomly chosen value from the distribution.
Factor or Divisor
When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product.
Quadratic Formula
X= -b (+/-) Sqrroot [(b^2) -4ac)] ----------------------------------- 2a
rate
a comparison of two quantities with different units. For example, number of steps to the amount of time.
b) temperature measurement
a) A unit used to measure angles
b) quadratic
a) ax + by = c, where a and b are not both zero
If a<b, then
a+c<b+c
a<b then a - b is positive or negative?
a-b is negative
a>b then a - b is positive or negative?
a-b is positive
a(b+c)
ab+ac
a(b-c)
ab-ac
straight angle
angle that measures exactly 180 degrees.
right angle
angle that measures exactly 90 degrees.
Pressure of liquid
density x gravity x height
1 is a divisor of
every number
What is the graph of f(x) shifted upward c units or spaces?
f(x) + c
What is the graph of f(x) shifted downward c units or spaces?
f(x) - c
Look at the graph for problem #7 on page 198. Write an equation for the graph in function notation
f(x) = 3x − 2
Look at the graph for problem #3 on page 198. Write an equation for the graph in function notation
f(x) = −x + 3
Solve h/3 = −2
h = − 6
radius
half the diameter
circle
infinite set of points equidistant from a centerpoint
To increase a number by x%
multiply by 1+x% i.e. 100 x (1+50%)=100x1.5=150
To decrease a number by x%
multiply by 1-x% i.e. 100 x (1-50%)=100x.5=50
distributive property
multiplying a number by a sum is the same as multiplying the number by each term in the sum and then adding the results.
Suppose you have a set of n objects, and you want to select k of them, but the order doesn't matter. What formula do you use to determine the number of combinations of n objects taken k at a time?
n! / (k!)(n-k)!
If you have a set of n objects, but you only want to order k of them, what formula do you use to determine the number of permutations?
n! / (n-k)!
Plot
to put a point on a coordinate grid.
evaluate
to replace the variable with a number and then simplify
Function notation
x represents the domain the f(x) represents the range
Write a compound inequality and solve: A number minus one is at most nine, or two times the number is at least twenty-four
x − 1 ≤ 9 or 2x ≥ 24 {x |x ≤ 10 or x ≥12}
Factor x^2 - xy + x.
x(x - y + 1)
Solve and graph the inequality x − (−5) > −2
{x | x > − 7} The graph is a number line with an open circle on negative seven and shading to the right.
Solve 6x + 12 < 8 + 8x
{x | x> 2}
Solve the compound inequality 4 < x + 6 and x + 6 < 5 Graph the inequality
{x | −2 < x < −1 The graph is a number line with an open circle on negative two and an open circle on negative one and shading between
Solve −5 − x/6 ≥ −9
{x |x ≤ 24 }
Solve 2(x + 3) ≥ 16
{x |x ≥ 5}
Solve and graph |x|< 3
{x |−3 < x < 3} Open circle on three and negative three with shading between
Look at problem 16 on page 176. You will see a graph. What is the slope of the graph?
−4/3
Simplify: (2xy)²(−3x²)(4y⁴)
−48x⁴y⁶
Simplify a monomial expression
Each base appears exactly once, there are no powers of powers, all fractions are in simplest form
Element
Each object or number in a set. Each entry in a matrix
Circumscribed
Each side of the polygon is tangent to the circle
fluently
Efficiently an accurately.
Set
Elements in a set do not repeat
Quadratic equation
Equation of the form ax² + bx + c = 0
three-dimensional
Existing in three dimensions; having length, width, and height.
What does scientific notation mean?
Expressing a number as the product of a decimal between 1 and 10, and a power of 10.
Quadratic expression
Expression in one variable with a degree of two
Factor and solve 2x² + 7x + 3 = 0
Factors are (2x + 1)(x + 3) = 0 Solutions are −3 and −1/2
Factor and solve 25p² − 16 = 0
Factors are (5x − 4)(5x + 4) = 0 Solutions are 4/5 and −4/5
Factor and solve 5d² − 22d + 8 = 0
Factors are (d − 4)(5d − 2) = 0 Solutions are 4 and 2/5
Factor and solve h²− 17h = −60
Factors are (h − 12) (h − 5) = 0 Solutions are 5 and 12
Factor and solve p² + 5p − 84 = 0
Factors are (p + 12)(p − 7) = 0 Solutions are −12 and 7
Solve an equation
Find the value of the variable that makes the equation true
Even integer
If an integer is divisible by 2, it is called an even integer.
Preimage
The original figure in a transformation
Sketch a quadratic equation that has two roots (solutions).
The parabola touches the x-axis at two points. See page 537
vertex (vertices)
The point at which two lines segments, lines, or rays, meet to form an angle.
Orthocenter of a triangle
The point of concurrency of the three altitudes of a triangle.
Incenter of a triangle
The point of concurrency of the three angle bisectors of a triangle
Circumcenter of a triangle
The point of concurrency of the three perpendicular bisectors of a triangle
greatest common divisor
The greatest number that divides into two or more numbers with no remainder.
How to recognize a # as a multiple of 4
The last 2 digits are a multiple of 4. (i.e 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4.)
How to recognize a # as a multiple of 4
The last 2 digits are a multiple of 4. (i.e 144. 44 is a multiple of 4, so 144 must also be a multiple of 4.)
Least Common Multiple
The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of both a and b.
Whole Numbers
The natural numbers and zero.
Union
The union of S and T is the set of all elements that are in S or T, or both, and is denoted by If sets S and T have no elements in common, they are called disjoint or mutually exclusive.
Polygonal Region
The union of a polygon and its interior
Output
The value generated for y.
Independent variable
The value of the variable that determines the output. The domain elements
Input
The value substituted for x.
x-intercept
The x-coordinate of the point where a graph crosses the x-axis
y-intercept
The y-coordinate of a point where a graph crosses the y-axis
Write the equations for two lines that are parallel.
There are many answers. An example would by y = 2x and y = 2x + 7.
Venn Diagram
To find total number of elements in all three sets: Add the numbers from each section together to get the total number of elements in all three sets. To find the total number of elements in one set: Add the numbers from each section that make up one circle to get the total number of elements in that one set. To find an unknown region: Label the unknown region that you want to find as "x." Then, label other unknown regions in terms of x. Then, add all regions of the graph and set it equal to the total number of elements in all three sets.
Describe how the graph of the function is related to the graph of f(x) = x² g(x) = x² + 9
Translates up 9
Similar triangles
Triangles that have the same shape but not necessarily the same size
What are congruent triangles?
Triangles with same measure and same side lengths.
Congruent Circles
Two circles with equal radii are called congruent circles.
Can you subtract 3sqrt4 from sqrt4?
Yes, like radicals can be added/subtracted.
pictograph
a graph that uses pictures or symbols to show data
common multiples
a number that is a multiple of all the given numbers (multiples that are common to two or more numbers)
constant
a number that stays the same in an equation, like 4. In y = 3x + 6, 6 is the constant.
Square Root
a number that when multiplied by itself gets you the number. For example, the square root of 16 is 4 because 4 times itself is 16.
coordinate
a number used to identify the location of a point: the x coordinate is written first
ordered pair
a pair of numbers that can be used to locate a point on a coordinate plane; (x,y)
degrees
a unit for measuring angles
Obtuse Angle
an angle with measure between 90 and 180 is called an obtuse angle.
permutation
an ordered arrangement of elements from a set
Two angles with the same measure are said to be
congruent
What is the graph of f(x) shifted left c units or spaces?
f(x + c)
The two rays that form an angle are called the _____ of the angle
sides
Graph 2x + 4y = 16 using x an y intercepts
x intercept is 8, and the y intercept is 4. See page 155 and 156 to check the work and graph
Graph 2x + y = −2 using the x and y intercepts
x intercept is −1 and the y intercept is −2
30 60 90
x, x(SR3), 2x
Look at the graph for problem #14 on page 218. What is the equation in slope intercept form?
y = −2 x + 3
Point-slope form
y − y1 = m(x ─ x1)
Solve −x/3 − 4 = 13
−51
In the equation, y = −2x − 6, the y intercept is _____________
−6
Solve |x + 1| = 5
−6 and 4
Scalene Triangle
3-sided polygon. No congruent sides
25^(1/2) or sqrt. 25 =
5 OR -5
octagon
8-sided polygon
What is a chord of a circle?
A chord is a line segment joining two points on a circle.
Perpendicular bisector
A line perpendicular to a segment at the segment's midpoint.
side
A line segment connected to other segments to form a polygon.
What is it called when a point is reflected to the quadrant opposite it (i.e. I to III or II to IV)?
A reflection about the origin.
Corollary
A theorem whose proof follows directly from another theorem.
Triangle
A three sided polygon
cone
A three-dimensional figure with one curved surface, one flat surface (usually a circle), one curved edge, and one vertex
Rotation
A transformation about a point P, also known as the center of rotation, such that each point and its image are the same distance from P. All of the angles with vertex P formed by a point and its image are congruent.
Area of a Parallelogram:
A=(base)(height)
Solve 3(x + 1) − 5 = 3x −2
All numbers
What are the integers?
All numbers multiples of 1.
What are the real numbers?
All the numbers on the number line (negative, rational, irrational, decimal, integer). All the numbers on the GRE are real. (-2, 1, .25, 1/2, pi)
Algebraic Expression
An algebraic expression has one or more variables and can be written as a single term or as a sum of terms.
Straight Angle
An angle that measures 180 degrees.
Multiplicative Inverse
A number and it's reciprocal.
Opposites
A positive number paired with a netagive number whose sum is zero. Also called additive inverses
System of equations
A set of equations with the same variables
Theorem
A statement that has been proven.
formula for area of a triangle
A=½bh
stem-and-leaf plot
A way to organize the numbers in a data set so that the numbers themselves make the display.
Composite number
A whole number greater than 1 that has more than 2 factors
Linear function
Ordered pairs that satisfy a linear equation
If E is certain
P(E) = 1/1 = 1
Probability of an Event
P(E) = number of favorable outcomes/total number of possible outcomes
Pyramid
Pyramid a polyhedron with a polygon base and triangular sides that all meet at a common vertex.
Pythagorean theorem
a² + b² = c²
Pythagorean theorem
a²+b²=c²
point slope form
The equation of a line in the form y-y1 = m (x-x1) where m represents the slope of the line and (x1, y1) is a known point on the line.
Write an equation for the nth term of the arithmetic sequence 7, 13, 19, 25 ...
The nth term = 6n + 1
Write an equation for the nth term of the arithmetic sequence 30, 26, 22, 18 ....
The nth term = −4n + 34
Describe the relationship between the graphs of x^2 and (1/2)x^2
The second graph is less steep.
least common multiple
The smallest number, greater than zero, found in all the list of multiples of two or more numbers.
Roots
The solutions of a quadratic equation
To multiply a number by 10^x
move the decimal point to the right x places
Find the value for n and name the property used: 7 + n = 7
n = 0 Additive identity
Find the value for n and name the property 5 · n · 2 = 0
n = 0 multiplication property of zero
Find the value for n and name the property used: 11· n = 11
n = 1 Multiplicative identity
Find the value of n and name the property used: 3 · 1/3 = n
n = 1 Multiplicative inverse or reciprocal
polygon
n-sided figure, closed, straight line segment, each segment only intersects exactly to other segments
An=A₁+(n-1)d
nth term
odd x odd =
odd
pattern
set of things in order that change in a regular way; Ex. Tile patterns, whose figure numbers and areas are represented with a table, a rule, or a graph
consecutive (adjacent) sides
sides that share an endpoint
representation of consecutive even/odd integers (depends what x is)
x, x+2, x+4, x+6, ...
(x^2)^4
x^(2(4)) =x^8 = (x^4)^2
x^4 + x^7 =
x^(4+7) = x^11
x^6 / x^3
x^(6-3) = x^3
Find the product (x + 9)²
x² + 18x + 81
Find the product (x − 7)(x + 5)
x² − 2x − 35
(x+y)²
x²+2xy+y²
factored binomial product of (x+y)²
x²+2xy+y²
(x-y)²
x²-2xy+y²
factored binomial product of (x-y)²
x²-2xy+y²
(x-y)(x+y)
x²-y²
binomial product of (x+y)(x-y)
x²-y²
Simplify. Assume no denominator is equal to zero. (3x³y)²/27x²
x⁴y²/3
Simplify x³ ∙ x²
x⁵
Simplify x⁹/x²
x⁷
The point slope form of a linear equation is ________
y - y1 = m(x − x1)
Graph x ≤ 3 and ≥ −2
Closed circle on 3 and closed circle on negative two with shading between
Sine
For an acute angle of a right triangle, the ratio of the leg opposite the acute angle to the measure of the hypotenuse
Negative exponent property
For any nonzero number a and any integer n, a−ⁿ= 1/aⁿ and 1/a−ⁿ = aⁿ
Pythagorean Theorem
For every right triangle, the area of the squares drawn off of the legs add up to the area of the square drawn off of the hypotenuse.
Quadratic formula
Formula used to find the solutions to a quadratic equation
If r, t, s & u are distinct, consecutive prime numbers, less than 31, which of the following could be an average of them (4, 4.25, 6, 9, 24, 22, 24)
4.25, 6, 22
Evaluate (4^3)^2
4096
40 < all primes<50
41, 43, 47
A company places a 6-symbol code on each product. The code consists of the letter T, followed by 3 numerical digits, and then 2 consonants (Y is a conson). How many codes are possible?
441000 = 1 * 10 * 10 * 10 * 21 * 21
In similar hexagons, the ratio of the areas is 16:25. What is the ratio of their corresponding sides?
4:5
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
4:9. The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides.
Simplify (a^2 + b)^2 - (a^2 - b)^2
4a^2(b)
What is the side length of an equilateral triangle with altitude 6?
4sqrt3. The triangle can be divided into two equal 30-60-90 triangles with side 6 as the side in which 6 = xsqrt3. So x =2sqrt3...
Find the GCF 40xy², 56x3y², 124x²y³
4xy²
Find the difference. (3x² − 7x + 5) − (−x² + 4x)
4x² − 11x + 5
hexagon
6-sided polygon
Inscribed
A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
face
A flat surface on a solid figure.
Plane
A flat surface that has infinite length and width, but no depth.
Plane
A flat surface that has no thickness and extends forever
slope intercept form
A form of a linear equation, y= mx +b, where m is the slope of the line and b is the y-intercept.
scientific notation
A form of writing as the product of a power of ten and a decimal number greater than or equal to one and less than ten.
quadrilateral
A four-sided polygon.
Fraction
A fraction is a number of the form a/b where a and b are integers and b≠0. The integer a is called the numerator of the fraction, and b is called the denominator.
Complex fraction
A fraction that has one or more fractions in the numerator or denominator
inverse function
A function in which two variables are inversely proportional.
quadratic function
A function with a second degree variable (x2)
What is a geometric sequence?
A geometric sequence has a first term that is not zero and each term after the first is found by multiplying the previous term by a nonero constant
bar graph
A graph that uses the height or length of rectangles to compare data
line graph
A graph used to show change over time with points connected by line segments.
scatter plot
A graph with one point for each item being measured.
combination
A group of items or events. Placing these items or events in a different order does not create a new combination.
population
A group of people (or objects or events) that fit a particular description.
Variable
A letter or symbol used to represent a value that can change.
Auxiliary line
A line drawn in a figure to aid in a proof
line of symmetry
A line that divides a figure into two congruent halves that are mirror images of each other.
Transversal
A line that intersects two coplanar lines at two different points.
axis of symmetry
A line which divides the graph of an equation into two congruent halves.
Segment bisector
A line, ray or segment that divides a segment into two congruent segments
regression line (line of best fit)
A line, segment, or ray drawn on a scatter plot to estimate the relationship between two sets of data.
Linear equation
A linear equation is an equation involving one or more variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1.
Lists
A list is like a finite set, having members that can all be listed, but with two differences. In a list, the members are ordered; that is, rearranging the members of a list makes it a different list. Thus, the terms "first element," "second element," etc., make sense in a list. Also, elements can be repeated in a list and the repetitions matter.
sample space
A list of all possible outcomes of an activity.
Point
A location in space that has no length, width, or depth.
inequality
A mathematical sentence that compares two amounts using the symbols; >, <, ≤, ≥, or ≠.
equation
A mathematical sentence with an equal sign.
Inequality
A mathematical statement that uses one of the following inequality signs is called an inequality.
mean (average)
A measure of center in a set of numbers, computed by adding the values in the list and then dividing by the number of values in the list.
Slope
A measure of the steepness of a line. Slope is the ratio of rise to run. For any two slopes on a line, the slope of the line is m = (y2− y1)/(x2 − x1)
Construction
A method of creating a figure that is considered to be mathematically precise
estimate
A number close to an exact amount; an estimate tells about how much or about how many.
Prime number
A number greater than 1, with only factors that are 1 and itself
composite number
A number greater than zero that has more than two different factors.
Coefficient
A number multiplied by a variable.
Square Root
A number multiplied by itself to form a product.
volume
A number of cubic units of space a solid figure takes up.
sample
A number of people, objects, or events chosen from a given population to represent the entire group.
mode
A number that appears most frequently in a set of numbers. There may be, one, more than one, or no mode.
rational number
A number that can be expressed as a ratio of two integers where the denominator is non-zero.
prime number
A number that has exactly two different positive factors, itself and 1.
common multiple
A number that is a multiple of two or more numbers.
positive number
A number that is greater than zero. Positive numbers are right of zero on a number line.
Coefficient
A number that is multiplied by variables is called the coefficient of a term.
Coordinate
A number used to identify the location of a point
Perfect Square
A number whose positive square root is a whole number.
Absolute Value
A number's distance from zero.
Monomial
A number, a variable, or the product of a number and one or more variables with nonnegative integer exponents
Term
A number, variable, or product or quotient of numbers and variables
Linear pair
A pair of adjacent angles whose noncommon sides are opposite rays
ordered pair
A pair of numbers that gives the coordinates of a point on a grid in this order (horizontal coordinate, vertical coordinate).
Line Segment
A part of a line between two points.
outlier
A piece of numerical data that is much smaller or larger than the rest of the data in a set.
Endpoint
A point at the end of a segment or the starting point of a ray
maximum of function
A point at which a function attains its greatest value.
vertex
A point where two or more straight lines meet.
Angle bisector
A ray that divides an angle into two congruent angles
Rational Numbers
A real number that can be written as a ratio of two integers. Rational numbers in decimal form are terminating or repeating.
Irrational numbers
A real number that cannot be expressed as the ratio of two integers.
Square
A rectangle with four congruent sides is called a square.
Rectangular Solid
A rectangular solid has six rectangular surfaces called faces. Adjacent faces are perpendicular to each other. Each line segment that is the intersection of two faces is called an edge, and each point at which the edges intersect is called a vertex. There are 12 edges and 8 vertices. The dimensions of a rectangular solid are the length the width w, and the height h.
Cube
A rectangular solid with six square faces is called a cube, in which case l=w=h.
axes
A reference line from which distances or angles are measured on a coordinate grid.
What transformation occurs if point C is reflected over the x-axis and then the y-axis?
A reflection about the axis.
property
A rule about numbers that is always true when you compute no matter which numbers you use.
transformation
A rule for moving every point in a plane figure to a new location.
Postulate
A statement that is accepted as true without proof
algorithm
A step-by-step method for computing or carrying out any mathematical procedure.
Line
A straight path that has no thickness and extends forever
Paragraph proof
A style of proof in which the statements and reasons are presented in paragraph form.
Two-column proof
A style of proof in which the statements are written in the left-hand column and the reasons are written in the right-hand column
numeral
A symbol or set of symbols representing or naming a number.
customary system
A system of measurement used in the United States. The system includes units for measuring length, capacity, weights, and temperature.
metric system
A system of measurement which units are based on tens.
Constant Term
A term that has no variable is called a constant term.
prism
A three-dimensional figure that has two congruent and parallel faces that are polygons. The rest of the faces are parallelograms.
cylinder
A three-dimensional figure with two circular bases that are parallel and congruent.
Reflection
A transformation across a line, called the line of reflection, such that the line of reflection is the perpendicular bisector of each segment joining each point and its image.
reflection
A transformation creating a mirror image of a figure on the opposite side of a line.
rotation
A transformation in which a figure is turned a given angle and direction around a point.
Reflection
A transformation that flips a figure over a line
Dilation
A transformation that makes a graph wider or narrower than the parent graph
Translation
A transformation that moves a figure up or down
dilation
A transformation that shrinks or enlarges a figure.
translation (slide)
A transformation that slides a figure a given distance in a given direction.
equilateral triangle
A triangle where all sides are congruent or equal. All angles are equal. Each angle measures 60 degrees
Right Triangle
A triangle with an interior right angle is called a right triangle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs.
Isosceles triangle
A triangle with at least two congruent sides
Isosceles Triangle
A triangle with at least two congruent sides is called an isosceles triangle. If a triangle has two congruent sides, then the angles opposite the two sides are congruent. The converse is also true.
Scalene triangle
A triangle with no congruent sides
scalene triangle
A triangle with no two congruent sides
Obtuse triangle
A triangle with one obtuse angle
Acute triangle
A triangle with three acute angles
Equiangular triangle
A triangle with three congruent angles
Equilateral triangle
A triangle with three congruent sides
Equilateral triangle
A triangle with three congruent sides is called an equilateral triangle. The measures of the three interior angles of such a triangle are also equal, and each measure is 60o.
Perfect square trinomial
A trinomial that is the square of a binomial
d=√(x₂-x₁)²+(y₂-y₁)²
Distance between two points on a graph
When asked to find the distance between 2 points on a graph use this formula...
Distance formula. Distance= Sqrareroot[((Xa-Xb)^2) + ((Ya-Yb)^2)]
d=rt
Distance travelled
skewed distribution
Distribution that shows bunching at one end and a long tail stretching out the other direction.
Coordinate system
Formed by the intersection of two number lines, the horizontal axis and the vertical axis
Angle
Formed by two rays that meet at an endpoint or formed by two intersecting lines. Measured in degrees.
Exponential function
Function in the form of y = ab× where a is not equal to zero and b > 0, and b not equal to 1
Nonlinear function
Function with a variable term other than 1
Scatter plot
Graphs in which two sets of data are plotted in ordered pairs in a coordinate plane
Is the equation y = 300(1.09)ⁿ exponential growth or decay
Growth
Equivalent equations
Have the same solution
congruent
Having exactly the same size and shape
two-dimensional
Having length and width.
equivalent
Having the same value.
vertex of quadratic equation
Highest or lowest point
Lateral face
Lateral face in a prism or a pyramid, a face that is not a base
Lateral surface
Lateral surface in a cylinder, the curved surface connecting the circular bases; in a cone, the curved surface that is not a base.
Length of an Arc
Length of an Arc = (central angle/360) x 2πr
Is the equation y = 1/2x + 5 linear, quadratic, or exponential?
Linear
Parallel lines
Lines in the same plane that do not intersect
Rational numbers
Numbers that can be expressed in the form a/b where a and b are integers and b is not equal to zero
quantitative relationships
Numbers that can be expressed or compared in a meaningful way.
Irrational number
Numbers that can not be expressed as a terminating or repeating decimal. Pi is an example.
irrational numbers
Numbers that cannot be written as a ratio of two integers. If you try to write an irrational number as a decimal, the digits never terminate and never repeat. ( EX √2 = 1.41421356...)
Irrational numbers
Numbers which are not rational but can be represented on the number line are called irrational numbers.
dividend
Numerator or number that is being divided, ex. the "a" in a/b. Dividend = (Divisor x Quotient) + remainder
1 is an
ODD number
y-axis
On a coordinate grid, the vertical axis.
outcome
One of the possible events in a probability situation.
Leg
One of the two sides of the right triangle that form the right angle
Square root
One of two equal factors of a number
Hertz (Hz)
One vibration per second
Graph x > 3 or x ≤ 0
Open circle on three and shading to the right. Closed circle on zero and shading to the left.
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . y = −2x and 2x + y = 3
Parallel
Parallel Lines Cut by a Transversal
Parallel Lines Cut by a Transversal
A quadrilateral where two diagonals bisect each other
Parallelogram
Grouping symbols
Parenthesis or brackets
Ray
Part of a line that starts at one point and goes infinitely in one direction.
Factor y² − 6y +8
(y − 2)(y − 4)
Use elimination to solve the system of equations: 2x − y = −1 and 3x − 2y = 1
(−3,−5)
dodecacon
12-sided polygon
1/8 in percent?
12.5%
The perimeter of a square is 48 inches. The length of its diagonal is:
12sqrt2
Legs 5, 12. Hypotenuse?
13
What is the surface area of a cylinder with radius 5 and height 8?
130pi
Evaluate if a = 12, b = 9, c = 4 a² + b − c²
137
A triangle is inscribed in a semi circle with legs 5 and 12. What is the circumfermence of the semicircle?
13pi / 2
Simplify the expression. If not possible write simplified 3x + 2(4y + 5x)
13x + 8y
Which is greater? 64^5 or 16^8
16^8 64^5 = (4^3)^5 = 4^15 16^8=(4^2)^8 = 4^16
If the 80th percentile of the measurements is 72degrees, about how many measurments are between 69 degrees and 72 degrees? Round your answer to the nearest tenth
18
Number of degrees in a triangle
180
If a pair of parallel lines is cut by a transversal that's not perpendicular, the sum of any acute angle and any obtuse angle is
180 Acute Angle an angle that is less than 90° Obtuse Angle:angle that is greater than 90° but less than 180°
What is the sum of the angles of a triangle?
180 degrees
The consecutive angles in a parallelogram equal
180°
the measure of a straight angle
180°
Circumference of a circle
2(pi)r
First 10 prime #s
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
First 10 prime #s
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself
Consecutive even integers
2, 4, 6, 8, ... Represented by n, n + 2, n + 4, n + 6
In a triangle where the two legs are 4 and 3, what is the value of a line directly intersecting the middle coming from the meeting point of the two legs?
2.4. We calculate the area (6) and then turn the triangle on its side and use x as the height to calculate again. (5x)/2=6
A brick with dimensions 10. 15 and 25 weighs 1.5 kg. A second brick (same density) has dimensions 12, 18, and 30. What is the weight of the second brick?
2.592 kg
Convert 66.66% to a fraction
2/3
Convert 40% to a fraction
2/5
Write an equation and solve. Twenty decreased by three times a number equals −10.
20 − 3x = −10 When the equation is solved, x = 10
If a=-1 and b=3, what is the value of (4(a^3)(b^2) - 12(a^2)(b^5)) / (16(a^3)(b^2))?
20.5
Solve the proportion. . . . . . . . . 9/(y + 1) = 18/54
26
Solve by completing the square. Round to the nearest tenth if necessary x² − 8x + 15 = 0
3 and 5
Two trains leave Chicago, one traveling east at 30 miles per hour and one traveling west at 40 miles per hour. When will the trains be 210 miles apart?
3 hours
Translate the sentence in to an equation. Three times the sum of g and h is 12.
3(g + h) = 12
x^2 = 9. What is the value of x?
3, -3
30 60 90
3, 4, 5
Consecutive odd integers
3, 5, 7, 9, ... Represented by n, n + 2, n + 4, n + 6
Triangle
3-sided polygon
Isosceles Triangle
3-sided polygon with at least 2 congruent sides.
Equilateral Triangle
3-sided polygon. 3 congruent sides. 3 60 degree angles.
Find the solution set if the replacement set is x = {0. 1/2, 1, 3/2, 2} for the equation 120 − 28x = 78
3/2
Convert 75% to a fraction
3/4
multiples of expressions (like if n+1 is a multiple of 5)
use a number line and add or subtract the number that the expression is a multiple of (like 5) from the expression (like n+1) to find other multiples
momentum (p)
velocity (v) x mass (kg)
The intersection of two sides of an angle is called the angle's
vertex
Y-axis
vertical line in a coordinate plane
Moments (N)
weight (N) x perpendicular distance (m)
instance of a conditional
when both the "if" and the "then" are true
product
when two or more numbers are multiplied
combine like terms
when you add or subtract like terms to simplify an expression
y-intercept
where the line intercepts the y-axis. Whatever y is when x is equal to 0. For example, in the equation y = 3x + 5 the y-intercept is 5.
Write an equation and solve. X plus 10 is equal to 3 times x.
x + 10 = 3x When the equation is solved, x = 5
Write the equation in standard form: y − 10 = −(x − 2)
x + y = 12
Solve the equation 0 = 4 − 2x
x = 2
Solve the equation for the variable indicated. 7x + 3y = m, for y
y = (m − 7x)/3
Suppose that the graph of f(x) is the result of stretching y=x + 5 away from the x-axis by a factor of 2. What is the new equation for the graph f(x)?
y = (x + 5)/2
Given the point (1, 3) and (−3, −5), write the equation in slope intercept form
y = 2x + 1
Suppose that the graph of f(x) is the result of sliding the graph of y=2x^2 down 3 units of spaces. What is the new equation?
y = 2x^2 - 3
The value of a new television depreciates by about 7% per year. You purchase a $3,000 TV. What is its value after 4 years. Write and solve an expontential equation to solve.
y = 3000(.93)⁴ In four years, the value of the TV is about $2244.16
Write an equation in slope intercept form for the line that passes through (−1, −2) and is parallel to 3x − y = 5.
y = 3x + 1
Write an equation in slope intercept form for the line that passes through (−2, 2) and is perpendicular to y = −1/3 x + 9.
y = 3x + 8
Which of the following equations is a direct variation equation? y = x + 2 or y = 3x
y = 3x because it is of the form y = kx
Write an equation in slope intercept form for the line that passes through (10, 5) and is perpendicular to 5x + 4y = 8.
y = 4/5x − 3
Given the point (1, 9) and the slope 4, write the equation in slope intercept form
y = 4x + 5
Write the equation in slope intercept form:. . . y + 2 = 4(x + 2)
y = 4x + 6
Exponential growth
y = a(1 + r)× where a is the initial amount, r is the rate of growth expressed as a decimal and t is time
Exponential decay
y = a(1 − r)× where a is the initial amount, r is the rate of decay expressed as a decimal and t is time
Direct variation
y = kx
What is the general equation for a direct variation equation?
y = kx
Slope-intercept form
y = mx + b
The slope intercept form of a linear equation is ____
y = mx + b
Write an equation in slope intercept form for the line that passes through (3, 2) and is parallel r to y = x + 5.
y = x − 1
Write the equation in slope intercept form: 2x + 4y = 12
y = −1/2x + 3
Look at the graph for problem #12 on page 218. What is the equation in slope intercept form?
y = −1/5 x + 1
Write the equation of the function that would translate the graph of x² over the x-axis, stretch it vertically by a factor of 2 and translate it down 3
y = −2x² − 3
Look at the graph for problem #34 on page 219. What is the equation in slope intercept form?
y = −4/7 x − 2
Look at the graph for problem #42 on page 235. Write the equation in point slope form
y − 7 = −4/3(x + 3)
In the equation, y = mx + b, the b stands for _____________
y-intercept
If y is directly proportional to x, what does it equal?
y/x is a constant
Solve the compound inequality x − 5 < − 4 or x − 5 ≥ 1 Graph the compound inequality.
{x | x < 1 or x ≥ 6 The graph is an open circle on 1 and shading to the left and a closed circle on 6 and shading to the right.
Solve the inequality −2x + 4 > −6 Graph the inequality.
{x | x < 5} The graph is a number line with an open circle on 5 and shading to the left.
Solve −8x − 3 < 18 − x
{x | x > − 3}
Solve x/6 ≤ 2
{x | x ≤ 12}
Solve and graph the inequality x + 12 ≥ 8
{x |x ≥ − 4} The graph is a number line with a closed circle on negative four and shading to the right.
Solve 3x + 17 < 4x Graph the inequality
{x|x > 17} The graph is a number line with an open circle on 17 and shading to the right.
Solve x + 11 > 16 Graph the inequality.
{x|x > 5} The graph of the number line is an open circle on five and shading to the right.
Plane geometry
Plane geometry is devoted primarily to the properties and relations of plane figures, such as angles, triangles, other polygons, and circles. The terms "point," "line," and "plane" are familiar intuitive concepts. A point has no size and is the simplest geometric figure. All geometric figures consist of points. A line is understood to be a straight line that extends in both directions without end. A plane can be thought of as a floor or a table top, except that a plane extends in all directions without end and has no thickness.
Parallel planes
Planes that do not intersect
Origin
Point (0, 0)
Coplanar
Point which lie on the same plane
y-y₁=m(x-x₁)
Point-slope form
Collinear Points
Points on the same line or line segment.
Collinear
Points which lie on the same line
If a is positive, aⁿ is
Positive
Slope of any line that goes up from left to right
Positive
∅ Is neither
Positive or Negative
Charle's Law
Pressure is constant
Positive integers that have exactly 2 positive divisors are
Prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23)
Probability
Probability is a way of describing uncertainty in numerical terms. In this section we review some of the terminology used in elementary probability theory. A probability experiment, also called a random experiment, is an experiment for which the result, or outcome, is uncertain.
Sphere
Sphere a three-dimensional figure with all points the same distance from the center.
ax+by=c
Standard form
Conjecture
Statement you believe to be true based on inductive reasoning
Logically equivalent statements
Statements that have the same truth value.
An Angle that's 180°
Straight Angle
Evaluate
Substitute numbers for the variables and simplify. (solve)
When dividing exponential #s with the same base, you do this to the exponents...
Subtract them. i.e (5^7)/(5^3)= 5^4
When dividing exponential #s with the same base, you do this to the exponents...
Subtract them. i.e (5^7)/(5^3)= 5^4
How to recognize a multiple of 6
Sum of digits is a multiple of 3 and the last digit is even.
Interior Angles of a Polygon
Sum of the angles = (n-2) x 180 (n is the number of sides)
Degree of a monomial
Sum of the exponents of all its variables
Trinomial
Sum or difference of three monomials
Binomial
Sum or difference of two monomials
How to find the sum of consecutive #s
Sum= (Average of Consecutive #s) * (# of terms in set)
Multiplication Principle
Suppose there are two choices to be made sequentially and that the second choice is independent of the first choice. Suppose also that there are k different possibilities for the first choice and m different possibilities for the second choice. The multiplication principle states that under those conditions, there are km different possibilities for the pair of choices.
How to find the average of consecutive #s
Take the average of the largest and smallest # in the set. (i.e. for the set of consecutive #s 5....172 would be (172+5)/(2)= 88.5)
Order of Operations
Tells you which operation to form first.
F=9/5C+32
Temperature conversion from Farenheit to Celsius
Boyle's Law
Temperature is constant
Like terms
Terms that contain the same variables raised to the same powers.
Like terms
Terms that contain the same variables, with corresponding variables having the same power
real numbers
The combined set of the rational and irrational numbers.
Vertex of an angle
The common endpoint of the sides of the angle
Dimensional analysis
Process of carrying units throughout a computation
Row reduction
Process of performing elementary row operations on an augmented matrix
Inductive reasoning
Processing of reasoning that a rule or statement is true because specific cases are true. You may use indicative reasoning to draw a conclusion from a pattern
Greatest common factor (GCF)
Product of the prime factors common to two or more integers
strategies
Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another.
Describe how you would graph y = 1/3 x + 2 using the slope and the y interept.
Put a dot on the y-axis on the number 2. Count a slope of 1/3 by going up 1 and right 3 or down 1 and left 3. Count the slope two or three times and then draw the line.
Sketch a graph that has a minimum at the point (4, −2) and has roots (or solutions) at 2 and 6
Put a point on the minimum of (4, −2) Put points on the x-axis on 2 and 6 and connect the parabola
Find hypotenuse of a right triangle given 2 side lengths
Pythagorean Theorem: h^2= (S1)^2 + (S2)^2
x=(-b±√b²-4ac)÷2a
Quadradic formula
Is the equation y = x² + 2x + 7 linear, quadratic or exponential?
Quadratic
Trigonometric ratio
Ratio that compares the side lengths of two sides of a right triangle
Terminating decimals
Rational numbers in decimal form that do not go on and on.
Repeating decimals
Rational numbers in decimal form that have a block of one or two digits that repeats continuously.
Graphs of Functions
The coordinate plane can be used for graphing functions. To graph a function in the xy-plane, you represent each input x and its corresponding output f(x) as a point (x, y), where y=f(x). In other words, you use the x-axis for the input and the y-axis for the output.
Coordinate plane
The coordinate plane is formed by the intersection of two perpendicular number lines called axes.
Natural Numbers
The counting numbers. (1,2,3,)
Decimal Number
The decimal number system is based on representing numbers using powers of 10. The place value of each digit corresponds to a power of 10.
Run
The difference in the x-values of two points on a line.
Rise
The difference in the y-values of two points on a line.
Term
The digits that need to be added or subtracted.
Absolute value
The distance a number is from zero on the number line
Circumference
The distance around a circle is called the circumference of the circle, which is analogous to the perimeter of a polygon. The ratio of the circumference C to the diameter d is the same for all circles and is denoted by the Greek letter p; that is,
Circumference
The distance around a circle.
perimeter
The distance around a figure.
Circumference
The distance around the circle
Length
The distance between the two endpoints of a segment
absolute value
The distance of a number from zero on the number line.
Domain of a Function
The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x. For the functions f and g defined above, the domain is the set of all real numbers. Sometimes the domain of the function is given explicitly and is restricted to a specific set of values of x.
Dividing by a number is the same as multiplying it by its
Reciprocal
Matrix
Rectangular arrangement of numbers in rows and columns enclosed in brackets
Extraneous solutions
Results that are not solutions to the original equation
In a rectangle, all angles are
Right
The small at the vertex of a right angle indicates that it is a
Right angle
A triangle with a 90 degree angle is a
Right triangle
Formula
Rule for relationships between certain quantities
Surface area of a rectangular solid
SA= 2( L*w + L*h + w*h)
Surface area of a sphere
SA= 4*pi*(r^3)
Vertex
The endpoint where two rays meet to form an angle.
Suppose y varies directly as x. Write a direct variation equation that relates x and y. Solve. If y = 7.5 when x = .5, find y when x = −0.3.
The equation is y = 15x. The value for y is −4.5
Suppose y varies directly as x. Write a direct variation equation that relates x and y. Then solve. If y = −4 when x = 2, find y when x = −6.
The equation is y = −2x. The value for y is 12
Exponent
The exponent tells how many times the base is used as a factor.
prime factorization
The expression of a number as a product of prime factors.
Radicand
The expression that is under the radical sign
Factorials
The factorial of n is the number of ways in which the n elements of a group can be ordered. The factorial of a number n represented by n! is the product of the natural numbers up to and including n. Example: 6! = 6 x 5 x 4 x 3 x 2 x 1
angle
The figure formed by two rays with the same endpoint.
Frequency Distributions
The frequency, or count, of a particular category or numerical value is the number of times that the category or value appears in the data. A frequency distribution is a table or graph that presents the categories or numerical values along with their associated frequencies. The relative frequency of a category or a numerical value is the associated frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals. A relative frequency distribution is a table or graph that presents the relative frequencies of the categories or numerical values.
square root
The number when multiplied by itself results in a given number.
Coefficient
The numerical factor in 6ab, the 6 is called the _________________-
What are the members or elements of a set?
The objects within a set.
Zeroes
The roots or x intercepts of a quadratic function
Equidistant
The same distance from two or more objects
Y-coordinate
The second number in an ordered pair; it lies on the vertical line.
Interior of an angle
The set of all points between the sides of an angle
Interior (of a polygon)
The set of all points inside a polygon
Exterior (of a polygon)
The set of all points outside a polygon
Exterior of an angle
The set of all points outside an angle
Sample Space
The set of all possible outcomes of a random experiment is called the sample space.
Interval
The set of all real numbers that are between m and n is called an interval.
What is the intersection of A and B?
The set of elements found in both A and B.
Intersection of two sets
The set of elements that are in both of the sets, denoted by the image
Union of two sets
The set of elements that are in either or both of the sets, denoted by image
What is the "union" of A and B?
The set of elements which can be found in either A or B.
How to recognize a # as a multiple of 3
The sum of the digits is a multiple of 3 (i.e. 45 ... 4 + 5 = 9 so the whole thing is a multiple of 3)
How to recognize a # as a multiple of 9
The sum of the digits is a multiple of 9.
How to recognize if a # is a multiple of 12
The sum of the digits it a multiple of 3 and the last two digits is a multiple of 4. (i.e 144: 1+4+4=9 which is a multiple of 3, and 44 is a multiple of 4, so 144 is a multiple of 12.)
Perimeter
The sum of the lengths of the sides of a polygon.
Perimeter
The sum of the side lengths of a closed plane figure
Surface Area
The surface area A of a rectangular solid is the sum of the areas of the six faces. A=2(lw+lh+wh).
Percent
The term percent means per hundred, or hundredths. Percents are ratios that are often used to represent parts of a whole, where the whole is considered as having 100 parts.
Set
The term set has been used informally in this review to mean a collection of objects that have some property, whether it is the collection of all positive integers, all points in a circular region, or all students in a school that have studied French. The objects of a set are called members or elements. Some sets are finite, which means that their members can be completely counted. Finite sets can, in principle, have all of their members listed, using curly brackets, such as the set of even digits {0, 2, 4, 6, 8}. Sets that are not finite are called infinite sets, such as the set of all integers. A set that has no members is called the empty set and is denoted by the symbol. A set with one or more members is called nonempty. If A and B are sets and all of the members of A are also members of B, then A is a subset of B.
Surface Area
The total area of the surface of a three-dimensional object
Evaluate. Express in scientific and standard form (4.9 X 10−³)/(4.0 X 10−⁵)
Scientific 1.96 X 10−⁷and standar 0.000000196
Evaluate. Express in scientific and standard form (4.8 X 10⁴)(6 X 10⁶)
Scientific 2.88 X 10¹¹ and standard 288,000,000,000
Formula for the area of a sector of a circle?
Sector area = (n/360) X (pi)r^2
Sketch a system of equations that has many solutions.
See the concept summary on page 333.
Sketch a system of equations that has no solution.
See the concept summary on page 333.
Solve the system of inequalities by graphing: 3x − y ≥ 2 and 3x − y < − 5
See the graph on page 383
Height
Segment from a vertex that forms a right angle with a line containing the base. Height may be a side of the triangle or in the interior or the exterior of the triangle,
Domain
Set of first numbers of the ordered pairs. The x values
Sequence
Set of numbers (terms) in a specific order
Relation
Set of ordered pairs
3-4-5 Right Triangle
Sides are in a ratio of 3:4:5
5-12-13 Right Triangle
Sides are in a ratio of 5:12:13
Corresponding sides (of polygons
Sides in the same position in polygons with an equal number of sides.
What form should you put lines in to determine if they are parallel, perpendicular, or neither?
Slope intercept form
Graph 3x − y < 2
Solve the equation for y (slope intercept form) Graph the line with a dotted line. Check a point to see what side to shade. See a graph for this inequality on page 315
Graph x + 5y ≤ 10
Solve the equation for y (slope intercept form) Graph the line with a solid line. Check a point to see what side to shade. See a graph for this inequality on page 316
Graph y − x = 4 using a table
Solve the equation for y. y = x + 4. Some points on the line are (0, 4), (1, 5), (−1, 3)
Graph y = 3ⁿ
Some points are (0, 0), (1, 3), (2, 9), (−1, 1/3), (−2, 1/9) See the graph on page 567
Graph y = −6x
Some points on the line are (0, 0), (1, −6),(−1, 6) See page 181 for a picture of the graph
Graph y = 1/3 x + 2 using a table
Some points on the line are (0, 2),(3, 3), (6, 4). Check page 156 to see the graph.
symbols
Something that represents something else.
Time Plots
Sometimes data are collected in order to observe changes in a variable over time. A time plot (sometimes called a time series) is a graphical display useful for showing changes in data collected at regular intervals of time. A time plot of a variable plots each observation corresponding to the time at which it was measured. A time plot uses a coordinate plane similar to a scatterplot, but the time is always on the horizontal axis, and the variable measured is always on the vertical axis.
Bivariate Data
Sometimes data are collected to study two different variables in the same population of individuals or objects. Such data are called bivariate data.
Base angles (of an isosceles triangle)
The two angles that have the base as a side.
Legs of an isosceles triangle
The two congruent sides of an isosceles triangle are called the legs.
Axes
The two number lines that form a coordinate plane.
Legs
The two sides of the right triangle that form the right angle. The sides of a right triangle that are not the hypotenuse
What is the set of elements which can be found in either A or B?
The union of A and B.
Dependent variable
The value of the variable that is dependent on the value of the independent variable. The range elements
Product of powers
To multiply two powers that have the same base, add their exponents
Scatterplots
To show the relationship between two numerical variables, the most useful type of graph is a scatterplot. In a scatterplot, the values of one variable appear on the horizontal axis of a rectangular coordinate system and the values of the other variable appear on the vertical axis. For each individual or object in the data, an ordered pair of numbers is collected, one number for each variable, and the pair is represented by a point in the coordinate system.
Equivalent equations
To solve an equation means to find the values of the variables that make the equation true, that is, the values that satisfy the equation. Two equations that have the same solutions are called equivalent equations.
Total clockwise moments
Total Anti-clockwise moments
Total momentum BEFORE collison
Total momentum AFTER collision
Translation
Transformation that shifts or slides every point of a figure or graph the same distance in the same direction.
T or F? Given d,e &f =/ 0, [(d^3)e(f^5)] / 2d(e^3) / [3(d^2)(e^3)(f^7)] / [6(e^5)(f^2)]?
True
True or false? 4.809 X 10^7 = .0004809 X 10^11
True
∅ is a multiple of
Two (∅×2=∅)
Linear Pair
Two adjacent angles formed by the intersection of two lines. These angles are supplementary (add to 180 degrees).
Adjacent angles
Two angles in the same plane with a common vertex and a common side, but no common interior points
Supplementary Angles
Two angles whose measures add to 180 degrees.
Complementary Angles
Two angles whose measures add to 90 degrees.
supplementary angles
Two angles whose measures have a sum of 180 degrees.
Supplementary angles
Two angles whose measures have a sum of 180°
complimentary angles
Two angles whose measures have a sum of 90 degrees.
Complementary angles
Two angles whose measures have a sum of 90°
What are "supplementary angles?"
Two angles whose sum is 180.
What are complementary angles?
Two angles whose sum is 90.
What is an isoceles triangle?
Two equal sides and two equal angles.
System of Equations
Two equations with the same variables are called a system of equations, and the equations in the system are called simultaneous equations.
dependent events
Two events in which the outcome of the first event affects the outcome of the second event.
independent events
Two events in which the outcome of the first event does not affect the outcome of the second event.
Equivalent Inequalities
Two inequalities that have the same solution set are called equivalent inequalities.
Perpendicular
Two lines in a plane that intersect at right angles
Parallel
Two lines in a plane that never meet
Parallel Lines
Two lines in the same plane that do not intersect are called parallel lines.
Vertical Angles
Two non-adjacent angles formed by two intersecting lines - across from each other. (<1 and <3 or <2 and <4)
Opposites
Two numbers are opposites if their sum is zero
multiplicative inverse
Two numbers whose product is 1 are multiplicative inverses of one another.
Multiplicative inverses
Two numbers whose product is one. Also called reciprocal
Concentric Circles
Two or more circles with the same center are called concentric circles.
Congruent
Two or more figures that are the same size and shape. Sides are equal in length and angles are equal in measure.
complementary events
Two or more mutually exclusive events that together cover all possible outcomes. The sum of the probabilities of complementary events is 1.
system of linear equations
Two or more related linear equations for which you seek a common solution.
Equally Likely Events
Two or more than two events are said to be equally likely if none of the events can occur in preference to the other.
Mutually Exclusive Events
Two or more than two events are said to be mutually exclusive, if they cannot occur at the same time.
Difference of perfect squares
Two perfect squares separated by a subtraction sign
Congruent polygons
Two polygons are congruent if and only if their corresponding angles and sides are congruent.
Opposite rays
Two rays that have a c common endpoint and form a line
angle
Two rays that share an endpoint.
Coordinate Geometry
Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a rectangular coordinate system, often called the xy-coordinate system or xy-plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by O.
Congruent segments
Two segments that have the same length
Translate the equation in to a sentence 2x + 10 = 26
Two times x plus ten equals twenty-six or the product of two and x increased by ten is twenty-six.
Congruent Triangles
Two triangles that have the same shape and size are called congruent triangles.
What is the slope of a vertical line?
Undefined, because we can't divide by 0.
Linear interpolation
Use a linear equation to predict values inside the range of the data
Vertical line test
Used to determine if a graph is a function
Conjugates
Used to form a rational number in a denominator that contains radicals and addition or subtraction
R
V / I
Refractive Index (n)
V1/V2
P
V2 / R
V1/t1
V2/t2
Volume of a sphere
V=(4/3)*pi*(r^3)
Volume of a rectangular box
V=L*w*h
formula for volume of a rectangular solid
V=l×w×h
formula for the volume of a cube
V=side³
zeros of function
Values of the variable for which the value of a function is zero. Also called roots of a function.
bh
Volume of a prism
1/3bh
Volume of a pyramid
4/3πr²
Volume of a sphere
Leading coefficient
When a polynomial is written in standard form, the coefficient of the first term
When does a function automatically have a restricted domain (2)?
When we need to avoid having a zero in the denominator or avoid taking the square root of a number.
plane
a flat surface that exceeds infinitely in all directions
expression
a group of symbols that make a mathematical statement 3 + 5 is an expression 2x - 3x + 6 is also an expression
What is a subset?
a grouping of the members within a set based on a shared characteristic.
term
a quantity in an expression separated by an addition or a subtraction symbol for example the expression "4 + x" has 2 terms (4 and x)
unit rate
a rate where the quantity in the denominator is one. For example, miles per hour, or $ per donut.
even x even =
even
odd x even =
even
Work (J)
force (N) x displacement (s)
perpendicular
forming right angles
velocity (waves)
frequency x wavelength (fλ)
Vertical
from up to down or down to up.
V1/V2
fλ1/ fλ2
compound statement
a sentence that combines two or more statements with some type of connective such as and, or, if-then
statement
a sentence that is either true or false, but not both
nonconvex sets
a set in which at least one segment that connects points within the set has points that lie outside the set
convex sets
a set of points in which every segment that connects points of the set lies entirely in the set
isosceles triangle
a triangle or trapezoid with two congruent sides
Distributive property
a(b + c) = ab + ac
b) point
a) A figure has line symmetry when it can be folded along a line so the two halves match exactly.
b) for function
a) The difference between the greatest and the least value in a set of data.
b) of a function
a) The inverse of a power. ax = b or = a, a is the xth root of b.
If a is inversely porportional to b, what does it equal?
ab=k (k is a constant)
F (N)
m (kg) x a (ms-2)
Solve −2m = 16
m = −8
Heat Capacity (C)
m x c ---> specific heat capacity
Potential Energy (Ep)
m x g x h
Slope given 2 points
m= (Y1-Y2)/(X1-X2)
E(h)
mCΔT
Weight (N)
mass (kg) x gravity (N kg -1)
Mechanical Efficient
mechanical advantage / velocity ratio
The objects in a set are called two names:
members or elements
obtuse angle
angle that is >90 degrees and <180 degrees.
Rational Numbers
any number that can be re-written as a ratio of two integers or a fraction.
Irrational Numbers
any number that cannot be re-written as a ratio of two integers or a fraction.
If a is negative and n is even then aⁿ is (positive or negative?)
aⁿ is positive
Write an equation to find the nth term of the sequence −2, 10, −50... and then use the equation to find the eleventh term
a₁₁ = −2 ∙ (−5)ⁿ⁻¹ The eleventh term is −19,531,250
The basic unit by which angles are measured
degree
perimeter
distance around a figure on a flat surface
When solving an inequality, flip the sign when you....
divide or multiply both sides by a NEGATIVE number
divisor
divisor Denominator or number you are dividing by, ex. the "b" in a/b.
Volume of a cube
edge³
Percent
Percentage equals part over whole times 100, or % = (part/whole)100 Part corresponds to "is" and Whole corresponds to "of," so % = (is/of)100 Overall: **** PART/WHOLE = IS/OF = %/100 ****
circumference
Perimeter of a circle.
2l+2w
Perimeter of a rectangle
4s
Perimeter of a square
a+b+c
Perimeter of a triangle
quadrants
one of the four regions into which the x- and y- axes divide the coordinate
Leg
one of the two shorter sides of a right triangle.
Circumference of a circle
pi(diameter)
Volume of a right circular cylinder
pi*(r^2)*h
integers
positive and negative numbers
factors
positive integers that can be evenly divided into the number (ie, there is no remainder)
table
represents numerical information by organizing it into columns and rows
n-gon
same as a polygon, where n is the number of sides
arithmetic sequence
sequence in which the difference between any two consecutive terms is the same ____________________________ An = A1 + (n - 1)d ____________________________ where An is the nth term in the sequence and A1 is the first term of the sequence and d is the difference between any two consecutive terms
geometric sequence/ exponential growth
sequence in which the ratio (r) between any two consecutive terms is the same _____________________________ An = A1 x (r ^ n-1) _____________________________ where An is the nth term in the sequence and A1 is the first term of the sequence and r is the ratio between any two consecutive terms
union of a set
set that includes ALL elements of the sets entering the union; the union of sets A and B is written A U B Example: if A = {1,2,3} and B = {4,5,6} , then A U B = {1,2,3,4,5,6}
If an inequality is multiplied or divided by a negative number....
the direction of the inequality is reversed.
What is a set with no members called?
the empty set, denoted by a circle with a diagonal through it.
dimensions
the lengths of parts of a figure that determine its shape and size; particularly the length, width, and height
exponent
the little number on the top to the right that tells you how many times to multiply the number by itself.
Area
the number of square units needed to cover a flat surface
base
the number on the bottom that is being multiplied by itself.
quotient
the number you get after you divide (analogous to what the product is after you multiply, but for division)
Slope
the rate of change shown on a line in a graph, it is calculated by the RISE divided by the RUN.
sum
the result of adding two or more numbers
Hypotenuse
the side opposite the right angle in a right triangle. The hypotenuse is the longest side of a right triangle.
LCM
the smallest multiple of two (or more) numbers (the smallest number that is a multiple of both)
A number is divisible by 3 if ...
the sum of its digits is divisible by 3.
decimal point
the symbol between the ones and the tenths in a decimal
isolate the variable
to get the variable all by itself. For example, isolate y in the equation y - 4x = 16 ----> Add 4x to both sides ----> y = 16 + 4x
Altitude of a triangle
Perpendicular segment form a vertex to the line containing the opposite side.
Pi is a ratio of what to what?
Pi is the ratio of a circle's circumference to its diameter.
Absolute value function
Piecewise linear function
Square of a difference
(a − b)² = a² − 2ab + b²
Solve the equation 2x(x − 3) = 0
0 and 3
C (unit J kg-¹ K-¹
E(h)/mΔT
What are the roots of the quadrinomial x^2 + 2x + 1?
The two xes after factoring.
1/6 in percent?
16.6666%
f(x) = 2x − 4 Find f(k + 1)
2k − 2
Set-builder notation
A concise way of writing a solution set
Additive Inverse
A number and its opposite.
Acute Angle
An angle with measure less than 90 is called an acute angle.
What is an arc of a circle?
An arc is a portion of a circumference of a circle.
C=πd
Circumference
Discriminant
In the Quadratic Formula, the expression under the radical sign, b² − 4ac
P and r are factors of 100. What is greater, pr or 100?
Indeterminable.
data
Information, especially numerical information
...-4, -3, -2, -1, 0, 1, 2, 3...
Integers
1, 2, 3, 4...
Natural numbers
Determine whether the equation is linear or not. If yes, write the equation in standard form y = 3x² + 1
No, because the x is raised to a power greater than one
Determine whether the equation is linear or not. If yes, write the equation in standard form xy = 6
No, because variables are multiplied together
E
Q x V
Closed half plane
The boundary line is included in the solution
sum
The result of addition.
Which is greater? 27^(-4) or 9^(-8)
27^(-4)
Find the volume of a cube whose length, width, and heighth have a measure of 3x⁵
27x¹⁵
If 8 schools are in a conference, how many games are played if each team plays each other exactly once?
28. n = 8, k = 2. n! / k!(n-k)!
How many 3-digit positive integers are even and do not contain the digit 4?
288 (8 * 9 * 4)
Reduce: 4.8 : 0.8 : 1.6
6 : 1 : 2
-3²
9
Evaluate 3& 2/7 / 1/3
9 & 6/7
Ratio of ages of Anna and Emma is 3:5 and of Emma and Nicolas is 3:5. What is the ratio of Anna to Nicholas' ages?
9 : 25
nonagon
9-sided polygon
The four angles around a point measure y, 2y, 35 and 55 respectively. What is the value of y?
90
Find the surface area of a cylinder with radius 3 and height 12.
90pi
Principal square root
The nonnegative square root
Find the product (5a − 2)(2a − 3)
10a² − 19a + 6
Find 5m²(2m³ −m)
10m⁵ − 5m³
10<all primes<20
11, 13, 17, 19
Evaluate 30 − 5 ∙ 4 + 2
12
From a box of 12 candles, you are to remove 5. How many different sets of 5 candles could you remove?
12! / 5!7! = 792
Use the distributive property to factor 24x + 48y = 0
12(x + 2y)
a^2 - b^2 =
(a - b)(a + b)
scale
...
symmetry
...
Write a verbal expression for 6x + 7
Six times x plus seven
contextualized problems
Solving real life situations using mathematics.
C=5/9(F-32)
Temperature conversion from Celsius to Farenheit
Open half plane
The boundary line is not included in the solution
Origin
The point of intersection.
bh
Volume of a cube
null set
a set that has no elements
diameter
distance across a circle through the center point
equivalent
having the same value
the slope of a line in y=mx+b
m
A polygon with four sides
quadrilateral
coefficient
the number multiplied by a variable in an equation or expression
vertex or vericies
the point where 2 lines meet
A number is divisible by 9 if...
the sum of digits is divisible by 9.
0^0
undefined
Slope intercept form
y = mx + b
(6sqrt3) x (2sqrt5) =
(6 x 2)(sqrt3 x sqrt5) = 12sqrt15
Factor the polynomial 9x² −3xy + 6x − 2y
(3x + 2)(3x− y)
Factor 16p² − 36
(4p − 6)(4p + 6)
If 10800 is invested at a simple interest rate of 4%, what is the value of the investment after 18 months?
$11,448
Hector invested $6000. Part was invested in account with 9% simple annual interest, and the rest in account with 7% simple annual interest. If he earned $490 in the first year of these investments, how much did he invest in each account?
$3,500 in the 9% and $2,500 in the 7%.
Use substitution to solve the system of equations: 3x − y = 4 and 2x − 3y = −9
(3, 5)
a^2 - 2ab + b^2
(a - b)^2
Simplify the expression [(b^2 - c^2) / (b - c)]
(b + c)
Time
(distance)/(rate) d/r
The sum of the measures of the n angles in a polygon with n sides
(n-2) x 180
Simplify the expression (p^2 - q^2)/ -5(q - p)
(p + q)/5
What is the maximum value for the function g(x) = (-2x^2) -1?
-1
-3³
-27
6w^2 - w - 15 = 0
-3/2 , 5/3
For what values should the domain be restricted for the function f(x) = sqrt(x + 8)
-8
X is the opposite of
-X
a) exponents
...
a) for algebraic ex: a + b
...
a) for data
...
a) line
...
a) linear
...
a) measuring device
...
b) A figure has point symmetry when it can be turned exactly 180º about a point and fit exactly on itself.
...
b) A system of marks at fixed intervals used in measurement or graphing.
...
b) The value which makes the equation equivalent to zero.
...
b) ax2 + by + c
...
b) unit of measure
...
c) The ratio of length used in a drawing, map, or model to the length of the object in reality.
...
expression
...
range
...
root
...
standard form for equations
...
The ratio of the areas of two similar polygons is ...
... the square of the ratios of the corresponding sides.
How many digits are there between the decimal point and the first even digit in the decimal equivalent of 1/[(2^8)(5^3)]
0
A cylinder has a surface area of 22pi. If the cylinder has a height of 10, what is the radius?
1
Probability of E not occurring:
1 - P(E)
Characteristics of a Parallelogram
1) 2 pairs of parallel sides 2) Opposite sides are equal 3) Opposite angles are equal 4) Consecutive angles add to 180 degrees
Characteristics of a Square
1) All sides are equal 2) Diagonals are equal 3) Area = sides²
Period (T)
1/f
sin c (critangle angle)
1/n
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. How many different people can get the three prizes?
10! / 3!(10-3)! = 120
What is the ratio of the sides of an isosceles right triangle?
1:1:sqrt2
What is the ratio of the sides of a 30-60-90 triangle?
1:sqrt3:2
What is the coefficient of the x^2 term in the product of (x + 1)(x + 2)(x -1)?
2
y = 2x The constant of variation is _________
2
Evaluate and write as a mixed number: 2/7 - 3/21 + 2 & 4/14
2 & 3/7
Evaluate and name the property used in each step 2 + 6(9− 3²) − 2
2 + 6 (9 − 9) − 2 (sub) . . . . . . . . . . . . . . . . . . . . 2 + 6(0) − 2 (sub) . . . . . . . . . . . . . . . . . . . . . . . . 2 + 0 − 2 (Mult prop of zero). . . . . .. . .. . . . .. . 2-2 (add iden). . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 (sub) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
State the dimensions of the matrix and identify the circled element for problem #12 on page 372
2 X 5 The circled three is in the second row and first column
Factor the monomial completely 42g²h
2 ∙ 3 ∙ 7 ∙ g ∙ g ∙h
Factor 8m − 6
2(4m − 3)
Nine coins are tossed simultaneously. In how many of the outcomes will the fourth coin tossed show heads?
2^9 / 2 = 256
Simplify 4sqrt21 X 5sqrt2 / 10sqrt7
2sqrt6
Wriite and solve in inequality: Twice a number minus 4 is less than three times the nunmber.
2x − 4 < 3x {x | x > − 4}
2⁵+2³
2⁸
In a Regular Polygon, the measure of each exterior angle
360/n
In any polygon, all external angles equal up to
360°
The sum of all angles around a point
360°
The sum of the angles in a quadrilateral is
360°
Find three consecutive odd integers whose sum is 117
37, 39, 41
3/8 in percent?
37.5%
30 60 90
3x, 4x, 5x
Simplify the expression. If not possible, write simplified 5x + 4 + 3x² − 3x
3x² + 2x + 4
Write 4 − x + 3x³ − 2x² in standard form and identify the leading coefficient
3x³ − 2x² −x + 4 The leading coefficient is 3
Legs: 3, 4. Hypotenuse?
5
30 60 90
5, 12, 13
Consecutive integers
5, 6, 7, 8, 9, ... Represented by n, n + 1, n + 2, n + 3, n + 4
pentagon
5-sided polygon
Express 0.000000058 in scientific notation
5.8 X 10⁻⁸
Express the number 0,00005816 in scientific notation
5.816 X 10⁻⁵
Convert 83.33% to a fraction
5/6
5 bakeries sell an average of 300 muffins per bakery per day. If 2 stop making muffins but the total muffins sold stays the same, what is the average of muffins per bakery sold among the remaining?
500
Simplify (−4xy)³(−2x²)³
512x⁹y³
What percent of 40 is 22?
55%
10^6 has how many zeroes?
6
How many sides does a hexagon have?
6
Find the solution set if the replacement set is x = {4, 5, 6, 7, 8} for the equation 5x − 9 = 26
7
In the equation y = 3x + 7, the y intercept is _______________
7
Convert 0.7% to a fraction.
7 / 1000
Find the next three terms of the arithmetic sequence 3.1, 4.1, 5.1, 6.1 ...
7.1, 8.1, 9.1
What is the third quartile of the following data set: 44, 58, 63, 63, 68, 70, 82
70
70 < all primes< 80
71, 73, 79
What is the ratio of the surface area of a cube with an edge of 10 to the surface area of a rectangular solid with dimensions 2, 4, and 6?
75:11
Find the degree of the polynomial 10x³y⁵
8
Solve the equation that follows 2(x + 2) =20
8
Write an algebraic expression for the sum of eight and the square of a number x
8 + x²
5/6 in percent?
83.333%
7/8 in percent?
87.5%
Find the GCF 88a³d, 40a²d², 32a²d
8a²d
In a Rectangle, each angles measures
90°
Angle
A figure formed by two rays with a common endpoint
The percent decrease of a quantity
= (actual decrease/Original amount) x 100%
If a lamp increases from $80 to $100, what is the percent increase?
= 25%. = (actual increase/original amount) x 100% = 20/80 x 100% = 1/4 x 100% = 25%
Circular Cylinder
A circular cylinder consists of two bases that are congruent circles and a lateral surface made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers of the two circles. The latter line segment is called the axis of the cylinder. A right circular cylinder is a circular cylinder whose axis is perpendicular to its bases.
Polygon
A closed figure formed by three or more line segments that intersect only at their endpoints.
Polygon
A closed plane figure formed by three or more segments such that each segment intersects exactly two other segments only at their endpoints and no two segments with a common endpoint are collinear
Augmented matrix
A coefficient matrix with an extra column containing the constant terms
b) for numerical ex: 6 + 4
A combination of variables, numbers and operation symbols that represents a mathematical relationship.
Bar Graphs
A commonly used graphical display for representing frequencies, or counts, is a bar graph, or bar chart. In a bar graph, rectangular bars are used to represent the categories of the data, and the height of each bar is proportional to the corresponding frequency or relative frequency. All of the bars are drawn with the same width, and the bars can be presented either vertically or horizontally. Bar graphs enable comparisons across several categories, making it easy to identify frequently and infrequently occurring categories.
Scalar
A constant that is multiplied by a matrix
Regular Polygon
A polygon in which all sides are congruent and all interior angles are congruent is called a regular polygon.
Polygons
A polygon is a closed figure formed by three or more line segments, called sides. Each side is joined to two other sides at its endpoints, and the endpoints are called vertices. The term "polygon" means "convex polygon," that is, a polygon in which the measure of each interior angle is less than 180o.
Regular Polygon
A polygon with equal angles and equal sides.
pyramid
A polyhedron whose base is a polygon and whose other bases are triangles that share a common vertex.
prime numbers
A positive integer greater than 1 that is only divisible by the number 1 and itself; in other words, it has only two whole number factors: itself and 1. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Permutation
A possible selection of a certain number of objects taken from a group with regard to order. (The number of ways a given set of things can be arranged, where order is considered to make a difference.) The permutation nPr is the number of subgroups of size r that can be taken from a set with n elements. It is calculated as follows: nPr = n! / (n-r)!
Combination
A possible selection of a certain number of objects taken from a group without regard to order. nCr is the number of unordered subgroups of size r that are selected from a set of size n. nCr = nPr / r! = n! / r!(n-r)! "OR" means you add "AND" means you multiply
unit
A precisely fixed quantity used for measure.
Prime Number
A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The integer 14 is not a prime number, since it has four positive divisors: 1, 2, 7, and 14. The integer 1 is not a prime number, and the integer 2 is the only prime number that is even.
rectangular prism
A prism with six rectangular faces.
triangular prism
A prism with triangular bases.
Indirect proof
A proof in which the statement to be proved is assumed to be false and a contradiction is shown.
Parallelogram
A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent.
Trapezoid
A quadrilateral in which two opposite sides are parallel is called a trapezoid.
trapezoid
A quadrilateral with exactly two parallel sides. OR A quadrilateral with one pair of parallel sides and one pair of sides that is not parallel.
parallelogram
A quadrilateral with two pairs of parallel and congruent sides.
constant
A quantity that always stays the same.
variable
A quantity that can have different values.
Radius
A radius is any line segment joining a point on the circle and the center of the circle.
Diameter
A segment that has endpoints on the circle and that passes through the center of the circle; also the length of that segment
Midsegment of a triangle
A segment that joins the midpoints of two sides of the triangle.
line
A set of connect points continuing without end in both directions.
arithmetic series (progression)
A set of numbers in which the difference between any two consecutive numbers is the same.
relation
A set of ordered pairs for which all x and y are related in the same way.
order of operations
A set of rules. It tells you the order in which to compute so that you will get the same answer than anyone else will get.
Pythagorean triple
A set of three nonzero whole numbers a, a, and c such that a squared plus b squared equals c squared.
What is a finite set?
A set with a number of elements which can be counted.
What is the empty set?
A set with no members, denoted by a circle with a diagonal through it.
Image
A shape that results from a transformation of a figure known as the preimage
polyhedron
A solid figure in which all the faces are polygons.
sphere
A solid figure made up of points that are the same distance from a point called the center.
Identity matrix
A square matrix that when multiplied by another matrix, equals that same matrix
Truth value
A statement can have a truth value of true (T) or false (F).
biconditional
A statement containing the words if and only if
experimental probability
A statement of probability based on the results of a series of trials.
prediction
A statement of what somebody thinks will happen in the future.
generalization
A statement or conclusion that is derived from and applies equally to a number of cases.
Conditional statement
A statement that can be written in the form "if p, then q," where p is the hypothesis and q is the conclusion
Biconditional statement
A statement that can be written in the form "p if and only if q."
Definition
A statement that describes a mathematical object and can be written as a true biconditional statement.
coordinate plane (Cartesian)
A two-dimensional system in which a location is described by its distances from two intersecting, usually perpendicular, straight lines called axes.
Degree
A unit of angle measure; one degree is 1/360 of a circle
Venn diagram
A useful way to represent two or three sets and their possible intersections and unions is a Venn diagram. In a Venn diagram, sets are represented by circular regions that overlap if they have elements in common but do not overlap if they are disjoint. Sometimes the circular regions are drawn inside a rectangular region, which represents a universal set, of which all other sets involved are subsets.
x-intercept
A value of y in an ordered pair describing the point at which a line or the graph of a function intersects the y-axis.
y-intercept
A value of y in an ordered pair describing the point at which a line or the graph of a function intersects the y-axis.
Constant
A value that does not change.
fraction
A way of describing a part of a whole or a group.
box (box-and-whisker) plot
A way to display a distribution of data values but using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.
dot/line plot
A way to display data values where each is shown as a dot or mark above a number line.
frequency chart
A way to display how often an item, number, or range of numbers occurs.
ordinal number
A whole number that names the position of an object in sequence ( first, second, third...)
Proof
An argument that uses logic to show that a conclusion is true
Base
The number used as a factor.
Explain the difference between an algebraic expression and a verbal expression.
Algebraic expression consists of numbers, variables, and arithmetic operations. Verbal expression consists of words.
Whole Numbers
All counting numbers and 0 form the set of whole numbers. Thus 0, 1, 2, 3, 4, 5 ... etc. are whose numbers. Clearly, every natural number is whole number and 0 is a whole number which is not a natural number.
Real numbers
All numbers that can be represented on the number line.
What are the rational numbers?
All numbers which can be expressed as a ratio of two integers. (All integers and fractions.) (-2, 1, .25, 1/2)
What are the irrational numbers?
All real numbers which can't be expressed as a ratio of two integers, positive and negative (pi, -sqrt3)
Functions
An algebraic expression in one variable can be used to define a function of that variable. Functions are usually denoted by letters such as f, g, and h.
Define an "expression".
An algebraic expression is a combination of one of more terms. Terms in an expression are separated by either addition or subtraction signs. (3xy, 4ab, -5cd, x^2 + x - 1)
rational expressions
An algebraic expression that can be written as a fraction whose numerator and denominator are polynomials.
Exterior angle
An angle formed by one side of the triangle and the extension of an adjacent side
Interior angle
An angle formed by two sides of a polygon with a common vertex
Obtuse Angle
An angle that measures between 90 and 180 degrees.
right angle
An angle that measures exactly 90º.
Acute angle
An angle that measures greater than 0° and less than 90°
Obtuse angle
An angle that measures greater than 90° and less than 180°
obtuse angle
An angle that measures greater than 90º and less than 180º.
acute angle
An angle that measures less than 90º.
What is an exterior angle?
An angle which is supplementary to an interior angle.
acute angle
An angle whose measure is between 0 degrees and 90 degrees
What is an arithmetic sequence?
An arithmetic sequence is a numerical pattern that increases or decreases at a constant rate called the common difference. The next term is found by adding the same positive or negative number to the preceding term.
linear equation
An equation in two variables whose graph in a coordinate plane is a straight line
counterexample
An example that shows a conjecture is false. Or for a conditional when the antecedent is true and the consequent is false.
Radical expression
An expression that contains a square root
vertical angles
Angles opposite one another at the intersection of two lines.
adjacent angles
Angles that have a common side and a common vertex (corner point).
Congruent angles
Angles that have the same measure
Corresponding angles
Angles that lie on the same side of the transversal t, and on the same sides of lines the lines r and s.
Same-side interior angles
Angles that lie on the same side of the transversal t, between lines r and s. Also called consecutive interior angles.
Adjacent Angles
Angles that share only one side and a vertex, but have no common interior points.
addend
Any number being added
whole number
Any of the numbers 0, 1, 2, 3, 4, and so on.
digit
Any one of the ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Event
Any particular set of outcomes is called an event.
classify
Categorize things or objects
Transformation
Changes the position or size of a figure
commutative property
Changing the order does not change the end result (applies to addition and multiplication)
How to find the circumference of a circle which circumscribes a square?
Circumference = Diameter(pi). Use pythagorean theorem to find the diagonal of the square (the diameter).
Included side
Common side of two consecutive angles in a polygon.
Ratio
Comparison of two numbers by division
Set builder notation
Concise way to write a solution set
Cone
Cone a three-dimensional figure with one vertex and one circular base.
Graphical Methods for Describing Data
Data can be organized and summarized using a variety of methods. Tables are commonly used, and there are many graphical and numerical methods as well. The appropriate type of representation for a collection of data depends in part on the nature of the data, such as whether the data are numerical or nonnumerical. In this section, we review some common graphical methods for describing and summarizing data.
Percent Change
Divide the change between the two values by the starting value, or percent change = (change/original)100 *Do not just add or subtract the percents together when a quantity is increased or decreased multiple times*
List the domain and the range for the following relation {(−2,−1), (3, 3), (4,3)}
Domain = {−2, 3, 4} Range = {−1, 3}
Power (P)
E or W / t (Js-1)
Proportion
Equation stating two ratios are equal
Radical equation
Equation that contains radicals with variables in the radicand
Linear equation
Equation that forms a line when it is graphed
Literal equation
Equation that involves several variables
Exhaustive Events
Events are said to be exhaustive if at least one of them must necessarily occur.
Independent Events
Events are said to be independent if the occurrence or non-occurrence of one event does not influence the occurrence or non-occurrence of the other event.
Rational Number
Every fraction with integers in the numerator and denominator is equivalent to a decimal that terminates or repeats. That is, every rational number can be expressed as a terminating or repeating decimal. The converse is also true; that is, every terminating or repeating decimal represents a rational number.
Prime Factorization
Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors. Such an expression is called a prime factorization.
∅ is a multiple of
Every number
Triangles
Every triangle has three sides and three interior angles. The measures of the interior angles add up to the length of each side must be less than the sum of the lengths of the other two sides.
Factor and solve x²− 16 = 0
Factors are (x − 4) (x + 4)) Solutions are 4 and −4
Whats the difference between factors and multiples?
Factors are few, multiples are many.
Solve the equation x² = 10x
Factors are x(x − 10) = 0 Solutions are 0 and 10
Evaluate
Find the value
Perform the indicated matrix operations for problem #28 on page 374. If the matrix does not exist, write impossible.
First row of matrix −9 4 second row of matrix −3 67 This is a 2 X 2 matrix
Write a verbal expression for the algebraic expression 5(x² +2)
Five times the quantity x squared plus two
Cosine
For an acute angle of a right triangle, the ratio of the leg adjacent to the acute angle to the measure of the hypotenuse
Tangent
For the acute angle of a right triangle, the ratio of the measure of the leg opposite the acute angle to the measure of the leg adjacent to the acute angle
Mapping
Illustrates how each element of the domain is paired with an element in the range
Subtract the two matrices for problem #18 on page 373. If the matrix does not exist write impossible
Impossible
domain
In a function, f(x), the possible values for x in the given situation.
Factors
In a multiplication expression, the numbers that are being multiplied
Hypotenuse
In a right triangle, the side opposite the right angle. The longest side of a right triangle
Exponent
Indicates the number of times the base is used as a factor
How many multiples does a given number have?
Infinite.
Perpendicular lines
Intersect to form 90 degree angles
Real numbers
Irrational and rational numbers together
Dependent Events
It implies that occurrence of one event affects the occurrence of the other event.
Binary Subtraction
It is easy to subtract a binary number from another binary number.
dimension
Measurement in one direction
Dispersion
Measures of dispersion indicate the degree of "spread" of the data. The most common statistics used as measures of dispersion are the range, the interquartile range, and the standard deviation.
Rationalize the denominator
Method to eliminate radicals from the denominator of a fraction
FOIL
Method to multiply two binomials
m=((x₁+x₂)÷2,(y₁+y₂)÷2)
Midpoint between two points on a graph
Polynomial
Monomial or the sum or difference of monomials
Constant
Monomial that is a real number
Permutations
More generally, suppose n objects are to be ordered from 1st to nth, and we want to count the number of ways the objects can be ordered. There are n choices for the first object, n-1 choices for the second object, n-2 choices for the third object, and so on, until there is only 1 choice for the nth object. Thus, applying the multiplication principle, the number of ways to order the n objects is equal to the product n(n-1)(n-2)....(3)(2)(1). Each order is called a permutation, and the product above is called the number of permutations of n objects. Because products of the form n(n-1)(n-2)...(3)(2)(1) occur frequently when counting objects, a special symbol n! called n factorial, is used to denote this product.
fact families
Number sentences that relate addition and subtraction or multiplication and division. Each number sentence in the fact family has the same numbers.
The length of one side of a triangle...
Must be greater than the difference and less than the sum of the lengths of the other two sides.
One is (a prime or not?)
NOT A PRIME
Point
Names a location and has no size; it is represented by a capital letter
Slope of any line that goes down as you move from left to right is
Negative
The product of odd number of negative numbers
Negative
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 2x + 5y = 15 and 3x + 5y = 15
Neither
Determine whether each pair of ratios is an equivalent ratio. 5/9, 7/11
No because when you cross multiply, 55 is not equal to 63
Solve 3(x − 6) = 3x
No solution
Solve the system of equations by graphing: . . . y = 2x + 3 and 3y = 6x − 6
No solution
Can you add sqrt 3 and sqrt 5?
No, only like radicals can be added.
Is the following an arithmetic sequence 1, 4, 9, 16...
No. There is not a common difference.
Alternate interior angles
Nonadjacent angles that lie on opposite sides of the transversal t, between lines r and s
7 divided by ∅
Null
Vertex angle (of an isosceles triangle)
The angle formed by the legs of an isosceles triangle.
Solve x² − 2x − 8 by graphing. What are the domain and range?
The axis of symmetry is 1. The vertex is (1, −9). Some points in the table may be (1, −9), (0, −8), (2, −8), (4, 0), (−2, 0) The solutions are 4 and −2. See the graph on page 537.Domain reals and range y ≥ −9
Solve x² + 6x + 8 = 0 by graphing and what are the domain and range.
The axis of symmetry is −3. The vertex is (−3,−1). Some points in the table may be (−3, −1), (−2, 0), (−4, 0), (−1, 3), (−5, 3) The solutions are − 4 and −2. Domain reals, Range y ≥ −1
probability
The chance of an event happening.
likelihood
The chance of something happening.
Union
The graph of a compound inequality containing or, the solution is a solution of either inequality, not necessarily both
parabola
The graph of a quadratic function. It is a symmetric curve.
Intersection
The graph of two inequalities that overlaps
capacity
The greatest amount that a container can hold.
Greatest Common Divisor
The greatest common divisor (or greatest common factor) of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b.
What is the "range" of a series of numbers?
The greatest value minus the smallest.
X-axis
The horizontal line that is perpendicular to the y-axis, and forms the coordinate plane.
Integers
The integers are the numbers 1, 2, 3, and so on, together with their negatives, -1, -2, -3, ... and 0. Thus, the set of integers is -3,-2,-1, 0, 1, 2, 3, ....
What is the set of elements found in both A and B?
The interesection of A and B.
Constant of variation
The k in the equation y = kx
minimum of function
The least value of a function.
Distance from a point to a line
The length of the perpendicular segment from the point to the line.
What is a major arc?
The longest arc between points A and B on a circle's diameter.
exponents
The number that tells how many equal factors.
area
The measure of the interior region of a two-dimensional figure or the surface of a three-dimensional figure.
slope
The measures of steepness of a line as you look at it from left to right. A numerical value for slope is found using two points on the line and dividing the change in y-value by the change in x-value.
median
The middle number when numbers are arranged from least to greatest. When the set has two middle numbers, the median is the mean of two middle numbers.
Negation
The negation of statement p is "not p," written ~p
Vertical angles
The nonadjacent angles formed by two intersecting lines
Area
The number of nonoverlapping unit squares of a given size that will exactly cover the interior of a plane figure
Dimension
The number of rows and columns in a matrix
powers
The number of times a number is repeated as a factor.
additive inverse
The opposite of a number. When a number is added to its additive inverse, the sum is zero.
Commutative property
The order you add or multiply numbers does not change their sum or product
Centroid of a triangle
The point of concurrency of the medians of a triangle.
What is the "solution" for a system of linear equations?
The point of intersection of the systems.
midpoint
The point of the segment that is equidistant from the segment's endpoints.
Midpoint
The point that divides a segment into two congruent segments
Standardization
The process of subtracting the mean from each value and then dividing the result by the standard deviation is called standardization. Standardization is a useful tool because for each data value, it provides a measure of position relative to the rest of the data independently of the variable for which the data was collected and the units of the variable.
Deductive reasoning
The process of using logic to draw conclusions
Product of a sum and a difference
The product of a + b and a − b is the square of a minus the square of b. (a − b)(a + b) = a² − b²
multiple
The product of a whole number and any other whole number.
perfect square
The product of an integer and itself.
Multiplicative property of zero
The product of any number and 0 is equal to 0
Multiplicative identity
The product of any number and 1 is equal to the number
monomial
The product of constants and variables.
What is true about the slopes of perpendicular lines?
The product of slopes of perpendicular lines is −1. (Slopes of perpendicular lines are opposite reciprocals)
rate of change
The ratio of change in one quantity to the corresponding change in another quantity. (see also slope)
Ratio
The ratio of one quantity to another is a way to express their relative sizes, often in the form of a fraction, where the first quantity is the numerator and the second quantity is the denominator.
Pi
The ratio of the circumference of a circle to its diameter, denoted by the Greek letter
product
The result of multiplication.
Real Numbers
The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line.
Hypotenuse
The side opposite the right angle in a right triangle
45-45-90 Isosceles Right Triangle
The sides are in a ratio of * x : x : √2
30-60-90 Right Triangle
The sides are in a ratio of x : x√3 : 2x
Perfect square
The square is a rational number
Radical sign
The square root symbol that is used to indicate a nonnegative square root
Standard Normal Distribution
The standard normal distribution is a normal distribution with a mean of 0 and standard deviation equal to 1. To transform a normal distribution with a mean of m and a standard deviation of d to a standard normal distribution, you standardize the values; that is, you subtract m from any observed value of the normal distribution and then divide the result by d.
Contrapositive
The statement formed by both exchanging and negating the hypothesis and conclusion of a conditional statement.
Converse
The statement formed by exchanging the hypothesis and conclusion of a conditional statement
antecedent
The statement that follows the "if"
consequent
The statement that follows the "then"
Trigonometry
The study of relationships among angles and sides of triangles
Additive identity
The sum of any number and zero is equal to the number
polynomial
The sum of monomials.
How to recognize a # as a multiple of 3
The sum of the digits is a multiple of 3
Axis of symmetry
The vertical line containing the vertex of a parabola
Y-axis
The vertical line that is perpendicular to the x-axis, and forms the coordinate plane.
Volume
The volume V of a rectangular solid is the product of its three dimensions, or V=lwh.
Associative property
The way you group 3 or more numbers when adding or multiplying does not change their sum or product
(event)=(favorable outcomes)/(Total outcomes)
Theoretical Probability
Write the equations for two lines that are perpendicular.
There are many answers. An example would by y = 2x and y = −1/2 x + 6. The product of the two slopes has to be −1
What is true about any horizontal line and a vertical line?
They are perpendicular
Binary Number System
This system has a base 2 and uses only 0 and 1; whereas the conventional decimal system having a base 10 uses 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Arc length
X = Measure of interior angle --------------- --------------------------------- Circumference of circle 360
Area of a sector
X = Measure of interior angle ---------------- ---------------------------------- Area of Circle 360
Right cone
a cone in which a line drawn from the base to the tip (vertex) passes through the center off the base
Can you simplify sqrt72?
Yes, because you can factor out a perfect square (36). Sqrt(36 x 2) = sqrt36 X sqrt2 = 6sqrt2.
Determine whether the equation is linear or not. If yes, write the equation in standard form y = 2 − 3x
Yes. 3x + y = 2
Segment
a part of a line between two endpoints on the line. For example, CD and PN are line segments.
Zero product property
f the product of two factors is zero, then at least one of the factors must be 0
What is the graph of f(x) shifted right c units or spaces?
f(x-c)
GCF
greatest common factor, or the largest common factor of two or more numbers (also called the GCD, or greatest common divisor) use the euclidean algorithm to solve http://m.youtube.com/watch?v=AJn843kplDw which uses long division and you keep on dividing the previous divisor by the remainder until the remainder is zero and then the divisor of that is the answer
Find the value of r so the line that passes through each pair of points has the given slope. (r, 3), (5, 9), m = 2
r = 2
Find the value of r so the line that passes through each pair of points has the given slope. (−4, 3)), (r, 5), m= 1/4
r = 4
sum of a geometric sequence
r = ratio between consecutive terms A1 = the first term An = the nth term Sn = the sum of the first n terms Sn = (A1 - A1 x (r^n)) / (1 - r )
Solve the quadratic equation ax^2 + bx + c= 0
x = [(-b)+/- (sqrt b^2 - 4ac)]/2a
Solve 2(4x + 3) + 2 = −4(x + 1)
x = −1
Solve 3.2x − 4.3 = 12.6x + 14.5
x = −2
Solve 18 − 4x = 42
x = −6
Consecutive integers
x, x+1, x+2
representation of consecutive integers
x, x+1, x+2, x+3, ...
Simplify (x⁷)⁴
x²⁸
$20,000 is invested at an interest rate of 5.2%. The interest is compounded quarterly. Write and evaluate an equation to determine how much money will be in the account in 10 years.
y = (20000)(1.013)⁴⁰ The amount would be $33528.01
#3 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?
• Sides with the same lengths are opposite angles with the same measure.
#2 What is an important property of a 30-60-90 triangle?
• The hypotenuse is twice the length of the shorter leg.
#1 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?
• The longest side is opposite the largest (biggest) angle.
#3 What is an important property of a 30-60-90 triangle?
• The ratio of the length of the three sides is x:x√3:2x
#3 What are the important properties of a 45-45-90 triangle?
• The ratio of the lengths of the three sides is x:x:x√2.
#2 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?
• The shortest side is opposite the smallest angle.
#1 What are the important properties of a 45-45-90 triangle?
• The triangle is a right triangle.
#1 What is an important property of a 30-60-90 triangle?
• The triangle is a right triangle.
Product of any number and ∅ is
∅
∅²
∅
The only number that is equal to its opposite
∅ ∅=∅
Probability of Event all cases
∅≤P(E)≤1
Solve the equation. Show work. x/5 + 6 = 2
− 20
Factor the monomial completely −38a²b
−1 ∙ 2 ∙ 19 ∙ a ∙ a ∙ b
Solve the equation. Show work. |5y − 2| = 7
−1, 9/5
In the equation y = −2x − 6, the slope is ________________
−2
Find the difference. (3a − 5)) − (5a + 1)
−2a − 6
Simplify. Assume no denominator is equal to zero. −15 w⁰u−¹/5u³
−3/u⁴
Find the sum (−4p² − p + 9) + (p²+ 3p −1)
−3p²+ 2p + 8
Solve by using the quadratic formula. Round to the nearest tenth if necessary x² + 5x = 6
−6, 1
Solve by completing the square. Round to the nearest tenth if necessary x²+ 6x = 7
−7 and 1
Solve the equation by using the quadratic formula. Round to the nearest tenth if necessary x² + 8x + 7 = 0
−7 and −1
Distance Formula
√( (x1 - x2)² + (y1 - y2)² )
