A lot of Algebra junk
Chapter 1 Section 2
*Applications and Problem Solving*
*3. If a = 0,*
*then −a = 0.*
Factoring can be done by...
... trial and error.
Example: 0/a
0.
What are the three types of Equations?
1. Contradiction - No solution. 2. Identity - There is a solution. 3. Conditional - Satisfied by some, but not by all values of the variable.
Addition Property of Equality
3x - 2 = 7 so 3x = 9
Inverse Property of Addition
8 + (-8) = 0
Inverse Property of Multiplication
8(1/8) = 1
factor 8m²+64
8(m²+8)
squaring a binomial
= to the square of the first term plus or minus twice the product of both terms plus the square of the second term (a+b)(sq) = (a)(sq) + 2ab +2b(sq) OR (a-b)(sq) = a(sq)-2ab+b(sq)
rectangle
A parallelogram with four right angles.
dodecagon
A twelve-sided polygon.
f(x) = X³ :
A vertical line that bends to the right, goes horizontal, then bends to the left, turning vertical again; this crosses at the (0,0)
area of a triangle
A= 1/2bh
area of a square
A= side²
Unordered lists:
Are sets of information; {2, 4, 8}
How does slope tell us the tilt? y/x
Because y/x = rise/run
Plan for Solving a Word Problem *Step 2*
Choose a variable and *use it with the given facts to* represent the unknowns *described in the problem.*
Polygon
Closed figure whose sides are line segments.
Empty Sets
Disjoint Sets { } or null.
inconsistent system
Has no solution
Classify: 5
Natural Number, Integer, Rational Number.
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 2x + 5y = 15 and 3x + 5y = 15
Neither
f(x) = X²:
Parabola pointing up; graphs at zero
P(n,r) = n!/(n-r)!
Permutations Formula
Coordinate Plane
Plane determined by the axes
orthogonality
Subspaces and orthogonality: A vector is only in a space (that is transposed) if it is orthogonal to every other vector in that space. WT is a subspace of Rn
c) (x⁴−9)/(x²+1)
Which function is shown in blue? a) (x⁴−9)/(x²+9) b) (x⁴−9)/(x²+2) c) (x⁴−9)/(x²+1)
Vertical line:
X = s ; slope undefined
category
a classification or grouping
Quadratic Function
an equation of the form x²
regarding
concerning about or with respect to
Element
each number in a matrix
reflections
flips a figure over a line
1-1
same images
substitution method
substituting 2nd equation into 1st equation
Interchange:
swap 2 equations
commutative property of multiplication
the order of the factors does not change the product a x b = b x a
principal
the original amount of money loaned or borrowed
represents
under or below
Integers
{...-2, -1, 0, 1, 2...}
factor a³+b³
(a+b)(a²-ab+b²)
factor a³-b³
(a-b)(a²+ab+b²)
Lesson *1-8*
*Number Lines*
Chapter 1 Section 3
*Numbers on a Line*
*4. The opposite of −a is*
*a; that is, −(−a) = a.*
Least Common Multiple
...
Rational Numbers
2/1, 1/3, (-1/4), 22/7, 0, 1.2, etc. Every integer is a rational number. Rational numbers can be expressed as decimals that either terminate (end) or repeat a sequence of digits.
2x² /x³
2/x
In the equation y = 3x + 7, the slope is _______________
3
Factor the polynomial completely: 3x³-81
3(x-3)(x²+3x+9)
Which of the following, based on the Descartes Rule of Signs, is the only possible classification of the roots of the function f(k) = -3k³ + 5k² - k + 4?
3, 1 positives; 0 negatives; 0, 2 imaginary
Pi
3.1415.... is an irrational number resulting from the ratio of a circle's circumference to its diameter
Simplify (¾)^-3
64/27
What is a Circle?
A Circle is a collection/set of points that are on a fixed distance (radius) from a fixed point (center).
pivot column
A column that contains a pivot position
*variable*
A symbol used to represent one or more numbers.
homogeneous
A system that can be written as Ax = 0; the x = 0 solution is a TRIVIAL solution
decagon
A ten-sided polygon.
Constant
A term that has no variable.
area of a rectangle
A= length x width
Intersection
All the elements that belong to both sets.
What is an Equation?
An Equation is a statement that two mathematical expressions are equal.
A linear system where two lines are parallel (no answer) would be what type of system?
An inconsistent system.
*real number*
Any number that is either positive, negative, or zero.
positive number
Any number greater than zero.
negative number
Any number less than zero.
What is Completing the Square?
As a Quadratic Expression: x^2 + kx + (k/2)2. As a Perfect Square Trinomial: x^2 + kx + (k/2)^2 = (x+k/2)^2.
Standard Form
Ax+By=c
What is the end behavior for f(x) = 3x⁴ - x² +1 ?
Both sides up.
What is the first step in graphing a linear inequality? y ≥ 1/2x - 3
Decide what type of line it will be. In this case it's a solid line because of the ≥ sign.
set
Designates a well-defined collection of numbers.
*opposite* *of a number*
Each of the numbers in a pair such as 6 and −6 or −2.5 and 2.5. Also called additive inverse.
addend
Each number in an addition problem.
What do parallel lines have in common?
Equal slopes. m¹ = m²
Complement
Everything that's NOT in that set.
FOIL method
F: product of FIRST terms O: product of OUTER terms I: product of INNER terms L: product of LAST terms; then combine like terms
What does FOIL stand for?
First, Outter, Inner, Last
What is the U (Union) Symbol?
Joins two pairs. You write this in between the inequality of the interval notation.
Stretch
Multiplys all y values by the same factor greater than 1
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
Positive Semidefinite
Q(x)0 for all x Eigenvalues are all non-negative
Classify: -1.2
Rational Number.
f(x) = ¼ X :
Stretch of a parabola
b = 4
This graph does NOT represent ax² -1/4, where a would be a fraction less than 1. This graph is of the form (x⁴ - a)/(x² + b) Find the integer, b. *HINT* Type "b= ..."
*Objective* 1-1
To simplyfy numerical expressions and evaluate algebraic expressions.
Parallel lines:
Two lines with equal slopes are...
straight angle
Two right angles form a straight angle.
Horizontal line:
Y = b ; slope 0
expression
a group of symbols that make a mathematical statement
squares
all sides are equal, opposite sides are parallel, all angles are 90 degrees
factor trinomials by grouping
ax(sq)+bx+c 1) find two numbers whose product is a*c and whose sum is b 2) rewrite bx, using the factors in step 1 3) factor by grouping ex: 3x(sq)+14x-5 step 1: 15&-1 step 2: 3x(sq)+14x-5 = 3x(sq)+15x-1x-5 step 3: 3x(x+5) -1(x+5) = (x+5) (3x-1)
Slope
change in y over the change in x (rise/run)
onto
consistent for any b; pivots in all rows
f ° g = :
f (g (x))
Linear Substitution
find out about linear substitution
zero factor theorem
if a & b are real numbers and if ab=0, then a=0 or b=0 ex: if (x+3)(x-1)=0, then x+3=0 or x-1=0
Equivalent Inequalities
inequalities that have the same solution
range
is the difference of the greatest value and the least value in a set of data.
Line Segment
part of a line
elapsed
passed
non-collinear
points that do not lie on the same line
prove
prove formally
Elementary row operation:
row replacement, row interchange, row scaling
Root
solutions to a quadratic equation
Line
straight line that has no width and no ends
When dividing two terms with the same base, you should _____ the exponents.
subtract
view
the act of looking or seeing or observing
Quotient
the result of a division problem
Product
the result of a multiplication problem
Difference
the result of a subtraction problem
Sum
the result of an addition problem
input
the x-value in a function
bisect
to cut or divide into two equal parts: to bisect an angle.
Solving a linear system means...
to find the solution set, i.e. the set of all possible solutions
Evaluate
to find the value of a numerical or algebraic EXPRESSION.
twice
two times
Brackets
used to group things to be done first, or show multiplication.
What is the Domain?
x.
Describe this inequality in interval notation: x < 5/3 ?
( -∞ , 5/3)
Associative Property of Multiplication
(ab)c = a(bc)
Perform the indicated operation: (7x - 3)²
49x² - 42x + 9
*equation*
A statement formed by placing an equals sign between two numerical or variable expressions.
Inequality
A statement that compares two quantities using <, >, ≤,≥, or ≠
scalene triangle
A triangle in which none of the sides are equal in length.
spectral decomposition
A=1u1uT1 +2u2uT2....... where u are the columns of P in A=PDPTwhere PT=P-1; this is the equation for matrix A where the spectrum of eigenvalues determine each piece
Universal Set
All the elements you can use.
approximately
Almost, but not exactly; more or less
What is the Slope-Intercept Form of an Equation of a Line?
Also known as a Linear Function.
orthonormal
An orthogonal set of unit vectors
corresponding
Angles or lines of 2 different polygons that are in the same position
Two ways to check when Dividing by Polynomials...
Dividend/Divisor = Quotient + Remainder/Divisor or Dividend = (Divisor)(Quotient) + (Remainder)
Characteristic (polynomials of square matrices)
Found by setting det(A-I)=0, a scalar equation that gives us roots of eigenvalues
How do we read? And how should I read a graph?
From left to right. Read a graph the same way.
integers
Includes negative numbers with a set of natural numbers.
Exponent
Indicates how many times the base gets multiplied by itself.
Power
Indicates the number of times a number is multiplied by itself.
How is a negative line viewed?
It runs downwards. m<0
Factor
Number multiplied by another factor to get a product in a multiplication problem.
Negative definite
Q(x)0 for all x≠0 All eigenvalues are all negative
Solving a linear system:
Replace the system by an equivalent system that is easier to solve
*evaluating* *a* *variable* *expression*
Replacing each variable in the expression by a given value and simplifying the result.
Null space
Set of all solution to Ax = 0
Column space
Set of all the linear combinations of the columns of A
How do you graph the line of a linear inequality? (Also the 2nd step) y ≥ 1/2x - 3
Set the inequality as an equation y ≥ 1/2x - 3 turns into y = 1/2x - 3, and plot the line.
f(x) = | X | - 3 :
Shifts V down three units
f(x) = ( X-3 )² :
Shifts parabola right 3 units
f(x) = X² + 2 :
Shifts parabola up two units
Conjugate Zeros Theorem is...
The Conjugate Zeros Theorem is if a Polynomial f(x) has only real coefficients and if a + bi is a zero of f(x), then the conjugate a - bi is also a zero f(x). Ex. f(x) = 1 + 2i is a zero. f(x) = 1 - 2i is also a zero.
The Fundamentals Theorem of Algebra is...
The Fundamentals Theorem of Algebra is a Polynomail of degree n with complex coefficients has a complex zero. Fact: A Polynomial of degree n has n zeros (counting multiplicity.)
rank
The dimension of the column space
additive inverse property
The sum of a number and its opposite is zero.
additive identity property
The sum of a number and zero is always that number.
Isolate the variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
f(x) = | X | :
V pointing up
Orthogonal Basis
Where all the vectors in the set are orthogonal to each other that forms a subspace; a.k.a where an orthogonal set that is also a subspace
Factors
Whole numbers that can be multiplied together to hind a product.
Scalene Triangle
a triangle in which all sides are different
proportion
an equation that states that two ratios are equal Ex: 1/2 = x/10
base
in an expression of the form x m the base is x
(-5)(2) = (2)(-5)
nm = mn
Negative Numbers
the negative counterparts of the positive real numbers
absolute value of zero
zero
In the equation y = −2x − 6, the slope is ________________
−2
Additive Identity Property
Addition Propery of Zero
nCr = n!/r!(n-r)!
Combinations Formula
hexagon
a six-sided polygon
Convex Polygon
any polygon without an indentation
Symbol of Inequality
the unequal sign which indicates that two quantities are not equal
common
to be expected
solutions
ways to solve problems
square
A rhombus with four right angles.
When multiplying binomials, what method do we use?
FOIL
Slope of a Line:
M= Y2 - Y1 / X2 - X1 , where X1 ╪ X2
Standerd Form
ax+by=c
Perimeter
the measure around
Circumference
the perimeter of a circle
Empty Set
when an equation has no solution
What is the Range?
y.
pentagon
A five-sided polygon.
Relation
A set of ordered pairs
Irrational Numbers
Any number that can't be written as a fraction.
How would you describe this inequality in interval notation: x ≥ 2 ?
[ 2 , ∞ )
zero exponent
a(0) = 1, as long as a is not 0
Rhombus
an equilateral parallelogram
Pythagorean Theorem
a² + b² = c²
per
by or through
international
concerning or belonging to all or at least two or more nations
variable expression
consists of numbers, variables, and operations
If you have more vectors than the number of entries in each of them, i.e. p>n then they are...
dependent
linear
designating or involving an equation whose terms are of the first degree
Dilation
enlargement or reduction of an image
What can a Linear Function Equation be written as?
f(x) = ax + b or f(x) = mx + b. The formula for a quadratic function is different from that of a linear function because it contains an x^2 term. Examples: f(x) = 3x^2 + 3x + 5 and g(x) = 5 - x^2 are not linear equations because of the ^2's.
(f/g)(x)= :
f(x)/g(x)
Constant function:
f(x)= mx + b , where m = 0
Linear Function formula:
f(x)=mx + b ; where m ╫ 0
What is Function Notation?
f(x)=y
power rule for exponents
if m and n are positive integers and a is a real number, then multiply exponents and keep the base a(m)(n)= a(mn)
product rule for exponents
if m and n are positive numbers, and a is a real number, then a(m) * a(n) = a(m+n) {add exponents but keep common base}
power of quotient rule
if n is a positive integer and a & c are real numbers, then (a/c)(n) = a(n)/c(n), and c does not equal 0
terms
in an expression are separated by addition and subtraction signs
perpendicular
intersecting at or forming right angles
interest
is the amount paid for borrowing or lending money
mode
is the value in a data set that occurs most often. If all values occur the same amount of times there is no mode for that set.
Translation
movement of a figure
Scaling:
multiply all terms in one equation by a nonzero constant
Scalar Multiplication
multiplying a matrix by a scalar
2/3 + 0 = 2/3
n + 0 = n
6 + 0 = 6
n + 0 = n
72 + 0 = 72
n + 0 = n
(½)(½) = (½)2
n ∙ n = n[sq]
2 ∙ 2 = 22
n ∙ n = n[sq]
3 ∙ 3 = 32
n ∙ n = n[sq]
Factorial Notation
n! = n × (n - 1) × (n - 2) × (n - 3) × ... × 3 × 2 × 1
.3/.3 = 1
n/n = 1
14/14 = 1
n/n = 1
5/8 ÷ 5/8 = 1
n/n = 1
9/9 = 1
n/n = 1
Irrational numbers
numbers that cannot be expressed in the form a/b, where a and b are integers and b =0.
Irrational Numbers
numbers that do not repeat or terminate
inverse
opposite in nature or effect or relation to another quantity
Orthonormal Vector
orthogonal vectors which have a length of 1(are unit vectors)
Grouping Symbols
parentheses ( ), brackets [ ], and braces { } that group parts of an expression.
graphing
picturing the solutions of inequalities on a number line. the picture is called the graph
Quadrilateral
polygon with 4 sides.
Zero is niether
positive nor negative.
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: g(x) = -x^5 + 2x⁴ + 3x² - 7x - 12
positive zeros: 0 or 2, negative zeros: 1, imaginary zeros: 2 or 4
experimental probability
probability based on what happens when an experiment is actually done
onto
range (-infinity, infinity)
secant
ratio of the hypotenuse to the adjacent side of a right-angled triangle
Best way to solve any system of linear equations is to...
row reduce the augmented matrix
population
statistics the entire aggregation of items from which samples can be drawn
translation
the act of changing in form or shape or appearance
volume
the amount of 3-dimensional space occupied by an object
tax rate
the amount of tax people are required to pay per unit of whatever is being taxed
diagonalizable matrix
the factorization of a matrix into three other matrices, PDP⁻¹, where D is a matrix containing the eigenvalues
narrowest
the field of vision is a cone-shaped area with its narrowest/widest end near the driver
Dividend
the first number in a division problem
Minuend
the first number in a subtraction problem
Domain
the first term in an ordered pair
degree of a polynomial
the greatest degree of any term of the polynomial
A linear system is consistent if and only if
the last column of the augmented matrix is not a pivot column, a consistent linear system has 1 solution precisely when it has no free variables
The leading entry of a nonzero is...
the leftmost nonzero entry
diameter
the length of a straight line passing through the center of a circle and connecting two points on the circumference
length
the linear extent in space from one end to the other
Coefficient
the number in front of the variable (ie: 5 in 5a)
Constant of Variation
the number k in equations of the form y=kx
area
the number of square units needed to cover a flat surface
opposite
the number that is on the other side of 0 and is exactly the same distance away from 0
Addend
the numbers added in an addition problem
Factor
the numbers multiplied in a multiplication problem
order of operations
the order in which operations in an expression to be evaluated are carried out. 1. parentheses 2. exponets 3. multiplication and divison 4. addition and subtraction
height
the perpendicular (90 degrees) distance from the base of a triangle to the opposite vertex
Point of Intersection
the place where two lines cross
intercept
the point at which a line intersects a coordinate axis
end point
the point in a titration at which a marked color change takes place
Spectrum (of a matrix)
the range or set of all eigenvalues of a square
odds
the ratio of the number of ways the event can occur to the number of ways the event cannot occur. Favorable over Unfavorable.
Divisor
the second number in a division problem
Subtrahend
the second number in a subtraction problem
Range
the second term in an ordered pair
degree of a term
the sum of exponents on the variables contained in the term
mean
the sum of the values in a data set divided by the number of values in the set
x=0 is the...
trivial, zero solution
Ordered Pair
two numbers to be plotted.
Independent Variable
variable that is changed in an experiment
x¹/³
³√x
What is the solution for |x - 5| < 0?
∅ The absolute value of a number can never be < 0. No solution.
In the equation, y = −2x − 6, the y intercept is _____________
−6
√5 √5 = :
√ 5 × 5
Are the inverse of each other:
√ and ( )²
Exponet rule:
√16+9 ╪ 4 + 3
vertex formula
(-b/2a, 4ac-b²/4a)
domain in interval notation f(x)=⁵√x-5 (for an odd index the root can be negetive or positive)
(-∞,∞)
At which value of x does f(x) = 2x³ - x² + 1 have a local minimum value?
(0,1)
Associative Property of Addition
(a + b) + c = a + (b + c)
factor a²-b²
(a+b)(a-b)
If x + 3 is a factor of x³ − x² − 17x − 15, what are the other factors?
(x + 1) and (x - 5)
greater than x, less than y in interval notation
(x, y)
Lesson *1-7*
*A Problem Solving Plan*
Title
*Algebra* Structure and Method Book 1
*substitution* *principle*
*An expression* *may be replaced by* *another expression* *that has the same value.*
*A = lw*
*Area of rectangle* = length of rectangle × width of rectangle
Plan for Solving a Word Problem *Step 5*
*Check your results* *with the words* *of the problem.* *Give the answer.*
Step 2
*Choose a variable and* *represent the unknows.*
*D = rt*
*Distance traveled* = rate × time traveled
Lesson *1-3*
*Equations*
Lesson *1-2*
*Grouping Symbols*
Chapter *1*
*Introduction to Algebra*
Step 1
*Read the problem carefully.*
Multiplication Property of -1
-1 * a = -a
Find all zeros of the polynomial function: g(x) = x³ - 2x² - x + 2
-1, 1, 2
Find all real zeros of the function: f(x) = x³ - 3x² - x +3
-1, 1, 3
Find all real zeros of the function: f(x) = x⁴ - 2x³ - 8x² + 8x + 16
-2, 2, 1 ± √5
Integers
-3, -2, -1, 0, 1, 2, 3, etc. These are the natural numbers, their additive inverses (negatives), and 0.
Find all zeros of the polynomial function: h(x) = 2x⁴ - 3x³ - 27x² + 62x - 24
-4, ½, 2, 3
Multiplication Property of -1
-5 = (-1)(5)
fundamental counting principle
...
permutations
...
A Projectile is...
... anything you can throw.
A is in Echelon from if it has 2 properties:
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it
A is in Reduced Echelon form if...
1) All nonzero rows have to lie above any row of all zeros 2) The leading entry of any nonzero row is to the right of the leading entry of any row above it 3) The leading entry of any nonzero row is 1 4) The entry is the only nonzero entry in the column
Row reduction algorithm
1) Locate the first pivot column/first pivot position 2) Locate a pivot, choose a nonzero entry in the pivot column to be the pivot, if necessary do row interchange to move the pivot to the pivot position, if necessary do row scaling to make the pivot equal to 1 3) Create zeros below the pivot use row replacement to make all entries below pivot equal to 0 4) Cover the row containing the pivot and any row above it, apply steps 1-3 to the remaining submatrix 5) Start from the right most pivot and then move upward and to the left, if a pivot does not equal 1, make it equal 1 with row scaling, use row replacement to create zeros above each pivot
factoring trinomials
1) use the form x(sq) + bx + c 2) factor out the GCF and then factor a trinomial of the form x(sq) + bx + c To factor ax(sq) + bx + c, try various combinations of factors of ax(sq) and c until a middle term of bx is obtained when checking
to solve quad expressions by factoring
1) write the equation in standard form: ax(sq)+bx+c=0 2) factor the quadratic 3)set each factor containing a variable = to 0 4) solve the equations 5)check in the original equation
Natural Numbers (Counting Numbers)
1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Whole numbers that are not negative.
Two rules when dealing with reflections and negatives are...
1. When negative is outside, it will make it look down. (Also the opposite.) 2. When negative is inside, it will make it look left. (Also the opposite.) * When dealing with square roots, when the negative is inside, the only values we can then add are those that are negative... which then make it a positive and is valid.
orthogonal component
1. x is in W' if x is perpendicular to every vector that spans W; 2. W' is a subspace of R^n
11x/2-x divided by 11/2-x:
11x/2-x × 2-x/11 = ?
Identity Property of Multiplication
1a = a
Systems of equations
2 or more equations with the same variable
2x⁻² /x³
2/x⁵
Expression
2x + 3
Subtraction Property of Equality
2x + 3 = 7 so 2x = 4
Equation
2x + 3 = 8
Using synthetic division: (2x³ - 25x² + 83x - 88) ÷ (x-8)
2x² - 9x + 11
Write the equation in standard form: y − 11 = 3(x − 2)
3x − y = −5
What is the greatest common mononial factor of 9x³y² + 15x²y - 6xy² ?
3xy
What is the complete factorization of 3x⁴ - 3x² ?
3x²(x-1)(x+1)
What is (3.2 x 10^5)(1.4 x 10^-2) written in scientific notation?
4.48 x 10³
Distributive Property
4x + 6 = 2(2x + 3)
Division Property of Equality
4x = 12 so x = 3
index of ⁵√x
5
Symmetric Property
5 = x so x = 5
Find all real zeros of the function: f(x) = x³ -6x² +4x - 24
6
ⁿ√60 where n=6:
60 1/6
In the equation y = 3x + 7, the y intercept is _______________
7
7c¹/⁶
7 ⁶√c
For what inequality symbols do you use parenthesis?
< and >
When graphing linear inequalities, for what symbols would you use a dotted line in the graph?
< and >
Linear graph in 2 variables:
A + By = C , where A or B ╪ 0
area of a circle
A = πr²
What are Complex Numbers?
A Complex Number is a number than can be written as a + bi. A is Real and B is Imaginary. >0 (is postive) - 2 distinct real sol. >| (bottom line) ( is zero) - Two solutons, but same. So 1. <0 (is negative) - No real solutions.
A Function is Neither if...
A Function is "Neither" if it is not odd nor even. Such as... + + +.
What is a Function?
A Function is a relation in which every element in a first set (domain) is paired to a unique element on a second set (range).
What is a Linear Equation?
A Linear Equation in one variable is an equation that can be written in the form ax+b=0. It must be equal to 1 (variable). It can not be squared or cubed unless they cancel out. Some may not even have variables, which it then is undefined.
What is a Transformation?
A Transformation is a shift or translations in the xy-plane.
A linear system where two lines cross (one answer) would be what type of system?
A consistent system.
Coordinate Plane
A coordinate system formed by the intersection of a horizontal number line, called the x-axis, and a vertical number line, called the y-axis.
When solving a linear system of equations, if the substitution or addition method resulted in 2 = 2, what would you have?
A dependent system, two lines on top of each other. ∞ Solutions
Variable
A letter or symbol used to represent a number such as x, y, a, b
Is this a linear equation or a linear function: y = 2x - 1 ?
A linear equation.
Is this a linear equation or a linear function: f(x) = 2x - 1 ?
A linear function
Basis
A linearly independent set in H that spans H; the pivot columns of A form a basis for A's column space
symmetric matrix
A matrix such that it equals its transpose; any two of its eigevectors from different eigenspaces ( made from different eigenvalues) are orthogonal; counting multiplicities, has as many eigenvalues as rows or columns;orthogonally diagonizable.
*positive number*
A number paired with a point on the positive side of a number line.
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
Prime
A number that has no other factors but itself and 1.
Composite
A number that has three or more whole number factors.
irrational number
A number with an infinite number of digits after the decimal point.
Term
A number, a variable, or the product of a number and a variable
ordered pair
A pair of numbers, (x, y), that indicate the position of a point on a coordinate plane.
ordered pair
A pair of numbers, x, y, that indicate the position of a point on a Cartesian plane.
equilateral polygon
A polygon in which all segments (or sides) are the same length.
regular polygon
A polygon in which all segments have the same lenght and all angles have the same measure.
concave polygon
A polygon with an indentation (or cave).
pivot position
A position in the original matrix that corresponds to a leading 1 in a reduced echelon matrix
Commutative Property
A property that states numbers can be added or multiplied in any order.
Associative Property
A property that states that numbers in addition of multiplication expressions can be grouped without affecting the value of the expression.
trapezoid
A quadrilateral that has exactly two parallel sides.
parallelogram
A quadrilateral that has two pairs of parallel sides.
What does Relation mean?
A relation is a set of ordered pairs.
triangle
A three-sided polygon.
dilation
A transformation that changes the size of an object, but not the shape.
obtuse triangle
A triangle in which one the angles measures more than 90 degrees.
equilateral triangle
A triangle in which the length of all sides are equal.
equiangular triangle
A triangle in which the measure of all angles are equal.
right triangle
A triangle that has a right angle.
isosceles triangle
A triangle that has at least two sides of equal length.
unit vector
A vector with a length of one (unit, hence the name)
The graph of x = b is?
A vertical line through b on the x axis. (Slope undefined)
diagram
A visual representation of data to help readers better understand relationships among data
Absolute Value is...
Absolute Value is the distance from zero; always positive.
Addition Property of Zero
Adding zero to a number is equal to the same number.
four basic math operations
Addition, subtraction, multiplication, and division
real number
All negative or positive numbers and zero.
linear equation
An equation that can be written as a1x1 + a2x2 + ... = b; a1, a2, etc. are real or complex numbers known in advance
rhombus
An equilateral parallelogram.
*variable* *expression*
An expression that contains a variable.
*numerical* *expression*
An expression that names a particular number; a *numeral.*
Transformation
An operation that moves or changes a geometric figure in some way to produce a new figure
positive real number
Any number that can be used to describe a physical distance greater than zero.
negative real number
Any number that can be used to describe the negative counterpart of a positive real number.
convex polygon
Any polygon that does not have an indentation. Most polygons you will study are convex polygons
*satisfy* *an open sentence*
Any solution of the sentence satisfies the sentence.
*solution* *of a sentence*
Any value of a variable that turns an open sentence into a true statement.
What three methods could be used to solve systems of linear equations? Equation¹: 2x + y = 6 Equation²: 3x - 2y = 16
By graphing, the substitution method, and the addition method.
Commutative Property
Change the order without changing the outcome.
Distributive Property
Distribute the the number to the other ones.
unit multipliers
Fractions used to change the units of a number.
What is the first step in this equation : 2|2x+1| -4 = 16 ?
Get rid of the variables outside of the absolute value.
congruent
Having the same size and shape
Addition Property of Equality
If a = b then a + c = b + c
absolute value
In reference to a number, the positive number that describes the distance on a number line of the graph of the number from the origin.
The Factor Theroem...
In the Factor Theroem, X-K is a factor of P(X) if and only if the remainder is P(K) = 0. (The remainder is 0). Ex. Decide if x-3 is a factor of x^3 - 2x +1 Use Synthetic, Linear or Remainder Theorem. P(X) = x^3 - 2x + 1 P(3) = 3^3 - 2(3) + 1 = 22 It's not 0, so x-3 is not a factor.
whole numbers
Includes the number zero with a set of natural numbers.
Origin
Intersection of the two axes.
Irrational Numbers
Irrational Numbers are numbers which are no rational numbers. They cannot be expressed as the ratio of two integers and has a decimal representation that does not terminate or repeat.
What is the Absolute Value Function?
It is V-Shaped and is represented by y=|x|. It cannot be represented by single linear function.
What is the end behavior for f(x) = -2x³ + x² - 2?
Left side up, right side down.
How do we view lines on the graph to get the correct slope?
Left to right
Principle Axes Theorem
Let A be a symmetric matrix; there is an orthogonal change of variable, x = Py, that transforms xTAx into yTDy with no cross-products
Long Division...
Make sure that everything is written in decreasing order of powers!
When Dividing Polynomials, you must make sure...
Make sure that everything is written in decreasing order of powers! Ex. 3x^3 - x^2 + 5 --> 3x^3 - x^2 + 0x + 5
f (x) :
Means plug a value for " X " in a formula " f "
What do you do when you have a negative denominator? ex b-2/-5
Move it up top. answer: 2-b/5 (Make sure to change signs)
Multiplicative Inverses (Reciprocals)
Multiplication Property of 1
(3³)² is the power of power property. It tells us to do what to the exponents?
Multiply exponents, so 3^5.
Multiplication Property of Zero
Multiplying 0 by a number is equal to 0.
Multiplication Property of 1
Multiplying 1 by a number is equal to the same number.
One to One function:
No two ordered pairs in a function have the same second componet (y-value)
Is ± ½ a possible rational solution of f(x)= - 3x³ - 11x² + 5x - 6?
No.
Order of Operations
Order to solve a problem. PEMDAS.
Malthusian Population Growth
P = Pµeⁿ° Where P is current population, Pµ is initial population, e is raised to the ()number in years times the °growth rate (birth rate - death rate)
How would you word the function (front) part of this equation? P(x) = 7.25x
P is a function of x
Order of Operations is...
PEMDAS Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction.
Two important things to remember in math are...
PEMDAS and to be careful with your signs!
What is the Vertex Form?
Parabola graph of f(x) = a (x-h)^2 + k with a not equal to 0; vertex (h,k). For the Vertex, h is always the opposite and k is what is displayed. This formula is known as the Standard Form of the Parabola.
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . y = −2x and 2x + y = 3
Parallel
Determine whether the lines are parallel, perpendicular, or neither: . . . . . . . . . . . . . . . . . 3x + 5y = 10 and 5x − 3y = −6
Perpendicular
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.
Indefinite
Q(x) assumes both positive and negative values Eigenvalues are both positive and negative
Classify: 13/7
Rational Number
*simplifying* *a* *numerical* *expression*
Replacing the expression by the simplest name for its value.
Plan for Solving a Word Problem *Step 3*
Reread the problem and write an equation *that represents* *relationships among the numbers* *in the problem.*
Reduced Echelon Form
Same as echelon form, except all leading entries are 1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
How do you find the y-intercept in a linear equation? ax + by = c
Set x = 0 to isolate the y.
How do you find the x-intercept in a linear equation? ax + by = c
Set y = 0 to isolate the x.
f(x) = | X - 3 | :
Shifts V three units right
Trace (of a square matrix)
Sum of the diagonal entries in a square matrix. Also important: It is the sum of the eigenvalues of A
What do you do when you divide a negative number in an inequality?
Switch the signs
*inequality symbols*
Symbols used to show the order of two real numbers. The symbol ≠ means "is not equal to."
A transformation T:R^n-R^m is linear if...
T respects the vector addition and scalar multiplication; i.e. T(u + v) = T(u) + T(v), T(cu)=cT(u)
What is the Average Rate of Change?
The Average Rate of Change of f from x^1 to x^2 is... y^2-y^1/x^2-x^1
What is the Range of a Function?
The Range of a Function f(x) is the set {y=f(x)}.
The Remainder Theroem...
The Remainder Theroem is when the remainder of dividing P(X) by X-K is P(K). You can also use Linear or Synthetic of course. Ex. f(2) = (2)^4 - 3(2)^2 - 4(2)^2 + 12(2)... Note: P(X) = (X-K) Q(X) + R If the remainder is 0, P(X) = (X-K) Q(X). Thus, (X-K) (The divisor) is a factor of P(X).
Property
The actions of numbers when combined.
Absolute Value
The distance a number is from the 0 on the number line
circumference
The distance around a circle
circumference
The distance around a circle.
Absolute Value
The distance away from zero. It is always positive.
radius
The distance from the center of a circle to any point on the circle.
greatest common factor
The largest factor that two or more numbers have in common.
Least Common Denominator
The least common multiple of the denominators of two or more fractions.
least common denominator
The least common multiple of the denominators of two or more fractions.
dimensions
The length, width, and height of an object being measured.
*positive integers*
The numbers 1, 2, 3, 4, and so on.
*values* *of* *a* *variable*
The numbers that can be represented by the variable.
factor
The numbers in a multiplication problem.
circumference
The perimeter of a circle.
*perimeter*
The perimeter of a plane figure is the distance around it.
point of intersection
The point where two lines cross.
*absolute value*
The positive number of any pair of opposite nonzero real numbers is the absolute value of each number in the pair. The absolute value of 0 is 0. The absolute value of a number a is denoted by |a|.
Rational Zero's Test is...
The possible Rational Zero's of f(x) are: factors of a^o / factors of a^n. Steps: 1. Consider the possible rational zeros. 2. Find the zero's of the quotient. 3. Write f(x) in factoral form.
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)
What is Mean?
The quotient of the sum of several quantities and their number; an average.
scale factor
The ratio of the lengths of two corresponding sides of two similar polygons
Base of a Power
The repeated factor in a power.
quotient
The result of a division problem.
product
The result of a multiplication problem.
sum
The result of an addition problem.
difference
The result of subtraction problem.
These two equations would be easiest to solve by what method Equation¹: y = -2x + 6 Equation²: 3x - 2y = 16?
The substitution method, substitue first equation in place of y in the second: 3x - 2(-2x + 6) = 16 and solve.
180 degrees
The sum of measures of the three angles in any triangle is 180 degrees.
perimeter
The sum of the lengths of the sides of a polygon
Change of Variable
To change: 1. orthogonally diagonalize matrix A 2. Make D a diagonal matrix with eigenvalues 3.substitute x^T Ax with (Py)^T A (Py) where x = Py 4.simplify to get λy² +λy²...
*solve* *an open sentence*
To find the solution set of the sentence.
*Objective* 1-3
To find solution sets of equations over a given domain.
*Objective* 1-1
To simplify numerical expressions and evaluate algebraic expressions.
*Objective* 1-4
To translate phrases into variable expressions.
*Objective* 1-6
To translate simple word problems into equations.
*Objective* 1-5
To translate word sentences into equations.
*Objective* 1-7
To use the five-step plan to solve word problems over a given domain.
*Objective* 1-9
To use opposites and absolute values.
Perpendicular lines:
Two lines whose slopes are negative reciprocal of each other are...
How do you do Intersection-of-Graph method on a Calculator?
Type in your results using Y=, Graph and then hit 2nd>Calc and Intersect to find the Intersection. Can also be done by hand.
Union symbol:
U
angles between real vectors
U·V=‖‖U‖‖ ‖‖V‖‖ cos (ϴ); ϴ=cos-1(U·V)/U·V
When doing Quadratic Equations...
When doing Quadratic Equations, you can separate the denominator. Such as... (-3/2) + or - (sqrt3/2) or (-3 + or - sqrt3)/(2)
Point slope form
Y-Y1=m(x-x1)
Slope-intercept form:
Y=mx + b
How do you find the equation of a line if you are given two points, but no y-intercept or slope? Such as: (4 , 3) (6 , -2)
You would have to find the slope first, x² - x¹/y² - y¹, and then use the point slope equation, y - y¹ = m(x - x¹)
What must you ALWAYS be careful with?
Your signs!
Associative Property of Multiplication
[(25)(16)](4) = (25)[(4)(16)]
ratio
a comparison of two numbers by division
repeating decimal
a decimal in which one or more digits repeat infinitely
Parameter
a determining or characteristic element; a factor that shapes the total outcome; a limit, boundary
tree diagram (probability)
a diagram used to show the total number of possible outcomes in an experiment
graph
a drawing illustrating the relations between certain quantities plotted with reference to a set of axes
Dependent Variable
a factor that can change in an experiment in response to changes in the independent variable
Absolute Value Function
a function written in the form y = /x/, and the graph is always in the shape of a v
point
a geometric element that has position but no extension
Scatter Plot
a graph of ordered pairs
Scatter Plot
a graph with points plotted to show a possible relationship between two sets of data.
variable
a letter used to represent one or more numbers
axis
a line about which a three-dimensional body or figure is symmetrical.
radius
a line segment from the center of a circle to any point on the circle (is also half the diameter)
Vertical Line Test
a method to determine if a graph is a function or not
centimeter
a metric unit of length equal to one hundredth of a meter
cubes
a number that is a whole number raised to the third power Ex. 8, 27, 64, 125, etc.
pyramid
a polyhedron having a polygonal base and triangular sides with a common vertex
Pythagorean triple
a set of three positive integers that work in the pythagorean theorem
rotation
a single complete turn axial or orbital
monomial
a single term made up of numbers and variables
Right Triangle
a triangle with a right angle
Mapping Diagram
a way to show a relation that links elements of the domain with cooresponding elements of the range
pendulum
a weight, hanging from a point, that swings an equal distance from side to side; often used in clocks
Inverse Property of Multiplication
a(1/a) = 1
Distributive Property
a(b + c) = ab + ac
a/b divided by c/d:
a/b × d/c = ?
Commutative Property of Multiplication
ab = ba
What is the Zero-Product Property?
ab=0 if and only if a=0 or b=0.
Replacement:
add a multiple of one equation to another
Equivalent expressions
algebraic expressions that have the same values for all values of variables
Real Numbers
all the positive real numbers plus zero
Additive Inverse
also known as the opposite
Multiplicative Inverse
also known as the reciprocal
Concave
an indentation
effect
an outward appearance
plane
an unbounded two-dimensional shape
Positive Real Numbers
any number that can be used to describe a physical distance greater than zero
meter
any of various measuring instruments for measuring a quantity
A solution to the linear system x_1, ..., x_n is...
any ordered n-tuple (S_1,...,S_n) of real numbers if you substitute S_1 for x_1, S_2 for x_2, ..., S_n for x_n, then every equation becomes a true relation
Least Square fit
approximating inconsistent systems of Ax = b; the smaller difference of ||b-Ax||, the better the approximation; ATAx(approx)=ATb(approx)
annual interest rate
apr, the percent of the principal you pay or earn a year
Column Row Expansion of AB
col1Arow1B + ...
identical
exactly alike
What will the solution of this absolute value equation look like |exp.| > a ?
exp. < -a or exp. > a
Increasing & Decreasing Functions
f increases on l if, whenever x^1<x^2, f(x^1)<f(x^2). f decreases on l if, whenever x^1<x^2, f(x^1)>f(x^2).
A Function is Odd if...
f(-x) = -f(x) (- + -) Always Symmetrical with respect to the origin.
A Function is Even if...
f(-x) = f(x) (+ - +) Always Symmetrical with respect to the y-axis.
vertex form
f(x) = a(x-h)² + k, where (h,k) is the vertex form of quadratic equation
Complete Factored Form...
f(x) = a^n (x-c)(x-b)(x-n)... Such as... 7x^3 - 21x^2 + 7x + 21 = 7(x+1)(x-1)(x-3)
Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -3, ± √2
f(x) = x³ + 3x² - 2x -6
Equivalent fractions
fractions that have the same value
equivalent fractions
fractions that have the same value and the same simplest form
no x value in the domain can be paired with more than one y value, vertical line test,
function
level
having a horizontal surface in which no part is higher or lower than another
h⁷/⁹ ÷ h⁵/⁹
h²/⁹
Imaginary Unit
i = √-1
spectral theorem
if A is symmetric, then it has real eigenvalues and can be orthogonally diagonalized
power
is a number made of repeated factors
base
is the number that is repeatedly multiplied in a power
reduced
made less in size or amount or degree
Tick Marks
marks used to denote sides and angles that have the same length and/or measure
formula
mathematics a standard procedure for solving a class of mathematical problems
(.9)(.9) = (.9)2
n ∙ n = n[sq]
(-5)1 = -5
nm = mn
(.4)(.6) = (.6)(.4)
nm = mn
4 ∙ 7 = 7 ∙ 4
nm = mn
Determinant
numbers in a square array enclosed by parallel lines
opposite of a number
on a number line, a number that is the same distance away from zero in the other direction
Function
one domain paired with exactly one range
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
(p³)⁴
p¹²
determine
shape or influence
Substitution Method
solve an equation and substitute it into another equation
quantity
something that has a magnitude and can be represented in mathematical expressions by a constant or a variable
like terms
terms that have identical variable parts raised to the same power
conclusion
the act of making up your mind about something
decrease
the amount by which something decreases
increase
the amount by which something increases
base
the bottom of a triangle
The free variable is...
the non-pivot column of the coefficient matrix
Divisor
the number by which a dividend is divided.
evaluate
to find the value of an expression
exceeds
to go beyond
creates
to make, form.
shown
to reveal
Straight Angle
two right angles that forms a straight line back to back
Equiangular Polygon
when all the angles of a polygon have the same measure
Regular Polygons
when all the sides of a polygon are the same length and all the angles are the same measure
Equilateral Polygon
when all the sides of a polygon have the same length
simplest form
when the GCF of the numerator and denominator is 1
Intersect
when two lines cross
Perpendicular
when two lines make square corners at the point of intersection
x-intercept
where a graph crosses the x-axis
in terms of
with regard to,with respect to
radicand of ⁵√x
x
Write the equation in standard form: y − 10 = −(x − 2)
x + y = 12
quadratic formula
x = -b ± √(b² - 4ac)/2a
What is the equation of the line through (0 , 2) that is perpendicular to the graph of the line: y = 1/2x - 4 ? (Think of the perpendicular rule)
y = -2/1x + 2 because (0 , 2) gives us the y intercept and -2/1 is the reciprocal of the line in the question.
What is the formula for slope-intercept form?
y = mx + b
y varies inversely as x (k is the constant)
y=k/x
Direct Variation
y=kx
y varies directly as x (k is the constant)
y=kx
⁴√x⁴
|x|
Opposites
4 and -4 -6 and 6
Quadrants
4 regions, roman numerals 1, 2, 3 and 4, in a counterclockwise order.
quadrilaterals
4 sided polygon
quadrilateral
A four-sided polygon.
Improper Fraction
A fraction whose numerator is greater than or equal to its denominator.
quadratic form
A function (Q()) where the vector inputed (x) = x^T Ax, where A is a symmetric matrix
Horizontal line test:
A function is one-to-one ONLYif every horizontal line intersects the graph at ONLY one point
Axes
A horixontal and vertical number line on a coordinate plane.
The graph of y = b is?
A horizontal line through b on the y axis. (Slope = 0)
Function
A relation which one element from the domain is paired with range
Translation
A transformation that "slides" each point of a figure the same distance in the same direction.
acute triangle
A triangle in which all angles measure less than 90 degrees.
f(x) = - √ x :
Graphs +x, -y to infinity
f(x) = √ -x :
Graphs -x, +y to infinity
f(x) = - √ -x :
Graphs -x, -y to infinity
f(x) = √ X :
Graphs as ½ of a horizontal parabola pointing towards ∞
These (dealing with i & Transformations) can be done by...
Hand and/or Calc.
consistent system
Has one or infinitely many solutions
X- Axis
Horizontal line
What is a Complex Conjugate?
If a + bi is given, then the Complex Conjugate is a - bi. Fact: (a + bi)(a - bi) = a^2 + b^2. The i disappears. Fact: (Conjugate Zeros Therom) - If a + bi is a solution at some polynomial eqn., then a - bi is also a solution.
Subtraction Property of Equality
If a = b then a - c = b - c
Division Property of Equality
If a = b then a/c = b/c
Multiplication Property of Equality
If a = b then ac = bc
Symmetric Property
If a = b then b = a
What is the Expression of sqrt(-a)?
If a > 0, then sqrt(-a) = i sqrt(a).
algebraic subtraction
If a and b are real numbers, then a - b = a +(-b) where -b is the opposite of b.
What is the Vertical Line Test?
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
dependent
If non-zero weights that satisfy the equation exist; if there are more vectors than there are entries
leading entry
Leftmost non-zero entry in a non-zero row
What is a Compound Inequality?
Make sure to isolate the variable in between. The solution set (in interval notation) of a compound inequality is always an interval of these: (a,b), [a,b], (a,b], [a,b).
natural (or counting) numbers
Numbers that are used to count objects or things.
*negative side*
On a horizontal number line, the side to the left of the origin.
*positive side*
On a horizontal number line, the side to the right of the origin.
Inverse Operations
Operations that undo each other.
x- coordinate
(Abscissa) 1st number in the pair
Associative Property of Addition
(27 + 38) + 12 = 27 + (38 +12)
modulus
(length of a complex number) z=√(a²+b²)=‖a,b‖
x²/³
(³√x)²
Properties of transposition
1. (A^T)^T = A; 2. (A+B)^T = A^T + B^T; 3. (rA)^T = r*A^T; 4. (AB)^T = B^T*A^T
Transpose Properties (of a matrix)
1. (A^t)^t = A 2. (AB)^t = Bt At 3. (cA)t = c (A)t
Ax = b
1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row
Distributive Property
2(x + 5) = 2x + 10
Real Numbers
2, -10, -131.3337, 1/3, etc. Real Numbers can be represented by decimal numbers. Real numbers include both the rational and irrational numbers.
Reciprocal
2/3 and 3/2 8 and 1/8
Commutative Property of Addition
3x + 2 + 5x = 3x + 5x +2
Identity Property of Addition
4x + 0 = 4x
If m=0 what type of line would be viewed on the graph?
A horizontal line.
Invertible Matrix Theorem (either all of them are true or all are false)
A is invertible; A is row equivalent to I; A has n pivot columns; Ax = 0 has only the trivial solution; The columns of A for a linearly independent set; The transformation x --> Ax is one to one; Ax = b has at least one solution for each b in R^n; The columns of A span R^n; x --> Ax maps R^n onto each R^m; there is an n x n matrix C such that CA = I; there is a matrix such that AD = I; A^T is invertible; The columns of A form a basis of R^n; Col A = R^n; dim Col A = n; rank A = n; Nul A = [0]; dim Nul A = 0
nonagon
A nine-sided polygon.
*negative number*
A number paired with a point on the negative side of a number line.
Function
A pairing of input and output values according to a specific rule.
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. aka "the Cartesian plane" after René Descartes
equiangular polygon
A polygon in which all angles have the same measure.
Constant
A term without a variable
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
one-to-one
A transformation that assigns a vector y in R^m for each x in R^n; there's a pivot in every column
Radioactive Decay Formula
Af=Ai(2)-t/h Where Af is Amount Final(at present) is equal to Amount Initial times two raised to the negative of the amount of time passed divided by the Halflife.
When solving a linear system of equations, if the substitution or addition method resulted in 2 ≠ 4, what system would you have?
An inconsistent system, parallel lines. No solution.
Rational Numbers
Any number that can be expressed as a fraction.
How can you Graph Quadratic Functions?
By hand or by using a calculator. A calculator is a good tool to check your answer by hand!
side
Each segment of a polygon.
Graph of the Ordered Pair
Graph that order pairs get plotted onto.
three dimensional
Having the dimensions of height, width, and depth.
independent
If only the trivial solution exists for a linear equation; the columns of A are independent if only the trivial solution exists
commutative property of addition
In a sum, you can add terms in any order, a + b = b + a
Fundamental question about a linear system:
Is the system consistent, i.e. does a solution exist, if it does, is this a unique solution?
How is a positive line viewed?
It runs upwards. m>0
equivalent ratios
Ratios that have the same value.
Plan for Solving a Word Problem *Step 1*
Read the problem carefully. *Decide what unknown numbers are asked for* *and* *what facts are known.* *Making a sketch may help.*
Shrink
Reduces y values by a factor between 1-0
f(X) = - | X + 7 | :
Reflection (V pointing down) shifted seven left
shown
Revealed
In the equation, y = mx + b, the m stands for _____________
Slope
How should you decide what side will be shaded in while graphing a linear inequality? y ≥ 1/2x - 3
Test it by putting in a point, typically (0 , 0) and see if it's true. 0 ≥ 1/2(0) -3 is 0 ≥ 3, which means true. Shade on the side of the point. If false shade on opposite side.
What is the Quadratic Function Equation?
The "a" in f(x) is called the Leading Coefficient.
Reflection about the "X" axis:
The - sign outside the arguement indicates what?
diameter
The radius of a circle times two.
Domain
The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
Reflection about the "Y" axis:
The sign inside the arguement indicates what?
Parent Function
The simplest function in a family; all functions in the family are transformations of it
How is slope defined?
The slope of a line measures the tilt of the line.
What is true about the slopes of parallel lines?
The slopes of parallel lines are the same
Length (complex and real vectors)
The square root of all the entries of the vector squared, if it is a vector with complex entries than you must use the absolute values of the complex numbers.
perpendicular
Two lines that make a square corner at the point of intersection.
Parentheses
Used to group things to be done first, or show multiplication.
Y- Axis
Vertical line
similar matrix
When two matrices A, B and another (invertible) matrix P satisfy A=P⁻¹BP
What is the basic Absolute Value Equation?
You can also have a number inside, such as |x-3| and the equal sign can be changed to something such as < or >.
equation
a mathematical sentence with an equal sign that shows that two expressions are equivalent
Equation
a mathematical statement in which two expressions are equal
inner product
a matrix product u^Tv or u . v where u and v are vectors; if U . V = 0, u and v are orthogonal
scientific notation
a method of writing very large or very small numbers by using powers of 10
value
a numerical quantity measured or assigned or computed
Rectangle
a parallelogram with four right angles
square unit
a square with sides one unit long.
Orthonormal Basis
a subspace spanned by an orthogonal set of unit vectors
chemical compound
a substance formed by the chemical combination of two or more elements in definite proportions
Numeral
a symbol which represents the idea of a particular number
constant
a term that has no variable and does not change
Acute Triangle
a triangle with angles smaller than 90 degrees
isosceles triangle
a triangle with at least two congruent sides
perfect square trinomial
a trinomial that is the square of some binomial: x(sq)+4x+4= (x+2)sq formulae: 1) a(sq)+2ab+b(sq)=(a+b)(sq); a(sq)-2(ab)+b(sq)=(a-b)(sq)
distributive property
a(b + c) = ab + ac an + ac = a(b+ c)
Distributive Property
a(b+c)=ab+ac, Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products.
sum or difference of cubes
a(cu)+b(cu) = (a+b)(a(sq) -ab+b(sq)) ex: y(cu)+8= y(cu)+2(cu) =(y+2) (y(sq)-2y+4) OR a(cu)-b(cu)= (a-b)(a(sq)+ab+b(sq) ex: 125z(cu)-1= (5z(cu)-1(cu)= (5z-1)(25z(sq)+5z+1)
difference of squares
a(sq)-b(sq) = (a+b) (a-b) ex: x(sq)-9=x(sq)-3(sq) = (x+3)(x-3)
sufficent
all that is needed, enough
independent event (probability)
an event that is not affected by another event.
dependent event (probability)
an event who's outcome does depend on the outcome of a previous event
radical expression
an expression that contains a square root
Number
an idea that represents quantity
transformation
assigns each vector x in R^n a vector T(x) in R^m
What is the standard form for a linear equation?
ax + by = c Both x and y are on the right side of the equation.
Discriminant
b²-4ac
promoting
calling attention to, advertising or publicizing
subtracting polynomials
change the signs of the terms being subtracted, then add ex: a-b = a+(-b)
associative property of multiplication
changing the grouping of factors will not change the product, (ab)c = a(bc)
associative property of addition
changing the grouping of terms will not change the sum, (a + b) + c = a + (b + c)
formed
clearly defined
Geometric Multiplicity
dimension of eigenspace (number of free variables)
Elimination Method
eliminate a variable by adding or subtracting the equations
Equivalent Equations
equations that have the same solution
Solving a two variable system amounts to...
finding the intersection of two lines
Transposition
flips rows and columns
standard form of quadratic equations
formula: ax(sq)+bx+c=0, with a not being 0 ex: x(sq)=16 in standard form is x(sq)-16=0 y=2y(sq)+5 in standard form is 2y(sq)+y-5=0
routine
found in the ordinary course of events
Polynomial Function
function represented by a monomial or a sum of monomials
g ° f = :
g(f(x))
Degree of a Polynomial
greatest degree of any term
(g⁴)²
g⁸
semi circle
half of a circle
Relative Maximum
highest point on a graph
regardless
in spite of everything
interval notation
instead of open circle, parenthesis are used. instead of closed circle, brackets are used. -infinity is all the numbers less than x to infinity.
median
is the middle value in a data set where the numbers are ordered least to greatest
circle
is the set of all points that are an equal distance from a point called the center
joined
joined
Parallel Lines
lines in the same plane that do not intersect and the distance between the lines is always the same
probability of an event
number of favorable outcomes divided by total number of possible outcomes
Algebraic Multiplicity
number of times an eigenvalue repeats. Or the number of times that a root appears in the characteristic polynomial.
Signed Numbers
numbers that are indicated with signs of postive or negative used in algebraic addition
Rational Numbers
numbers that stop and can be written as fractions
quadrant
one of four sections into which the coordinate plane is divided
If v_1,v_2,..,v_pER^n then the set {v_1,v_2,v_p} is dependent if and only if
one of the vectors is a linear combination of the preceding ones
Inverse Operation
operations that undo each other, such as addition and subtraction
modeled
resembling sculpture
v_1, v_2, ..., v_p are dependent if and only if...
the system has >= 1 free variable
The homogeneous system Ax=0 has a nontrivial solution if and only if...
the system has infinite many solutions, the system has at least one free variable
Synthetic Substitution
the use of synthetic division to evaluate a polynomial
GCF of list of common variables raised to powers
the variable raised to the smallest exponent in the list.
The basic variables are...
the variables corresponding to the pivot column in the coefficient matrix
height
the vertical dimension of extension
v_1...v_p are linear dependent if...
there is a nontrivial linear relation between them; i.e. fi we can find some scalars c_1, c_2, ..., c_p, not all zeros, where c_1v_1 + c_2v_2 +...+c_pv_p = 0
Two matrices A and B are row equivalent if...
there is a sequence of row operations that transform one into another
v_1...v_p are linear independent if...
there is no non-trivial linear relation between them, i.e. has only the trivial solution
Two linear systems are equivalent if...
they have the same solution set
theoretical probability
what should occur in a probability experiment...an experiment is not actually done
Leontief input-output model
x = Cx + d
A zero of a function is also the _____ on a graph?
x-intercept
Multiplication Property of Equality
x/5 = 4 so x = 20
Quadratic Equation
x=(-b±√b²-4ac)/2a
Completing the Square Example...
x^2 + 4x + 5 1. x^2 + 4x ____ + 5 2. 4/2. ALWAYS divide by 2! Then square the answer; 2^2 = 4. Now add AND THEN ALSO subtract it. 3. (x^2 + 4x + 4) + 5 - 4 4. Now make it a perfect square! (x + 2)^2 + 1. This is the Standard Form - f(x) = a(x - h)^2 + k. 1(x+2)^2+1; 1 is a, 2 is h and 1 is k. Vertex is (-2,1) and it is going Upwards.
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
Look at the graph for problem #14 on page 218. What is the equation in slope intercept form?
y = −2 x + 3
Look at the graph for problem #34 on page 219. What is the equation in slope intercept form?
y = −4/7 x − 2
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)
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
Slope Intercept Form
y= mx + b where "m=slope" and "b=y-intercept"
Intersection symbol:
∩
intersection, includes only the numbers in common from 2 sets, x<5, x>2 →includes only numbers less than 5 but greater than 2 (2, 5)
∩
union, includes all possibilities from both sets, x<5, x>2 →includes all real numbers(⁻∞, ∞)
∪
For what inequality symbols do you use brackets?
≤ and ≥
When graphing linear inequalities, for what symbols would you use a solid line in the graph?
≤ and ≥
What is an Ordered Pair?
(x, y)
Orthogonal Projection
turning a vector into two other vectors r that sum up to it requires an orthogonal projection; Finding the weights for the linear combination: C1 =Y·U1/U1·U1 , given that y=c1v1+c2v2...
Gram-Schmidt Process
used for making orthonormal/orthogonal bases; The process, which makes a set of vectors {x1..xn} into an orthogonal basis {v1...vn} 1. v1 = x1 2. v2 =x1- (x2·v1/v1·v1)v1 3. v3 = x3- (x3·v1/v1·v1)v1-x3·v2/v2·v2)v2 (etc.)
milligrams
used to measure the mass of very small objects one thousandths
A linear equation is...
variables x_1, x_2,...,x_n, any equation that can be written in the form: a_1x_1+a_2x_2+...+a_nx_n=b; a_1, a_2, ..., a_n are real numbers
⁵√x⁵
x
Given the point (1, 3) and (−3, −5), write the equation in slope intercept form
y = 2x + 1
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
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
The slope intercept form of a linear equation is ____
y = mx + b
slope intercept form
y = mx + b
How do we calculate slope from two given points? (x¹,y¹) (x²,y²)
y² - y¹/x² - x¹ to get the slope. (rise/run)
complex number
z=a+bi where i=√-1
conjugate (of complex numbers)
z=a-bi, etc.
Whole Numbers
{0, 1, 2...}
Counting Numbers
{1, 2, 3...}
What is the solution for |x - 5| ≤ 0?
{5} Absolute value of a number can never be < 0 but it can be 0 when x =5. |5 - 5| = 0
How would you express this inequality in a solution set: -4 < t ≤ 5/3 ?
{t| -4 < t ≤ 5/3}
Determine the domain and write your answer in set-builder notation. v+3/v-5
{v|v≠5}
How would you describe this inequality in a solution set: x < 5 ?
{x|x<5}
Like Terms
Terms that have the same variable and the same exponent.
Positive definite
Q(x)0 for all x≠0 All eigenvalues are all positive
Q is directly proportional to y and inversely proportional to y (k is the constant)
Q=kx/y
What is the solution for |x - 5| ≥ 0?
R. Any real number, because the absolute value of a number is always ≥ 0.
What is the degree of the polynomial h(t) = -8t² + 5 - 3t³?
The degree is 3
Real Zero's of a Polynomial facts...
Real Zero's of a Polynomial = x-intercept. No x-intercept means no real zeros.
f(x) = - X³ :
Reflection of X³
vertex
The endpoint of each segment in a polygon.
pi
The exact numberof times the diameter of circle will go around the circle, which is approximately 3.14.
dividend
The first number in a division problem.
minuend
The first number in a subtraction problem.
*domain* *of a variable*
The given set of numbers that the variable may represent.
*negative integers*
The numbers −1, −2, −3, −4, and so on.
divisor
The second number in a division problem.
subtrahend
The second number in a subtraction problem.
*solution set* *of an open sentence*
The set of all solutions of the sentence.
*integers*
The set consisting of the positive integers, the negative integers, and zero.
*whole numbers*
The set consisting of zero and all the positive integers.
Integers
The set of all whole numbers, their opposites, and 0.
decimal system
The system of numeration that is used to designate numbers.
surface area
The total area of the 2-dimensional surfaces that make up a 3-dimensional object.
f(x) = | X + 2 | :
Shifts V two units left
f(X) = | X | + 1 :
Shifts V up one unit
f(x) = X³ - 5 :
Shifts graph down 5 units
f(x) = ( X + 3 ) ³ :
Shifts graph left 3 units
f(x) = ( X - 6 ) ³ :
Shifts graph right 6 units
f(x) = X³ + 3 :
Shifts graph up 3 units
f(x) = X² - 3 :
Shifts parabola down three units
f(x) = ( X + 2 ) ² :
Shifts parabola left 2 units
f(x) = 4 X² :
Shrink of a parabola
What form should you put lines in to determine if they are parallel, perpendicular, or neither?
Slope intercept form
Slope Formula
Slope=m= Y2 - Y1 / X2 - X1
*sides* *of an equation*
The two expressions joined by the equals sign.
What is the difference in slopes for perpendicular lines?
They have negative reciprocals. ex. Line¹: m= 2/4, would mean Line²: m= -4/2
Solve for a variable
To get a variable alone on one side of an equation or inequality in order to solve the equation or inequality
*Objective* 1-8
To graph real numbers on a number line and to compare real numbers.
*Objective* 1-2
To simplify expressions with and without grouping symbols.
If x-2 is a factor of a polynomial f(x), is it true that f(2) = 0 ?
True.
How do you do a Quadratic Equation by Calculator?
Use Y^1 & Y^2 in your Calc. Hit Graph, and then use the Intersection.
tables
Used to arrange text in columns and rows
zero
Used to describe a physical distance of no magnitude or an empty set.
Cauchy-Schwarz inequality
We use this to find the angle between two real vectors in terms of approximations; (find out more!)
orthogonal diagonalization (of a real symmetric matrix)
When an orthogonal matrix P with a diagonalized matrix D where A=PDP⁻¹=PDP^T (P⁻¹=P^T)
Linear Function
a function in which the graph of the solutions forms a line
quadratic equations
a function that has the variable raised to the second power
Trend Line
a line that approximates the relationship between the data sets of a scatter plot
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself
function
a mathematical relation such that each element of one set is associated with at least one element of another set
Eigenvalue
a scalar, nonzero solution to Ax=λx;Basically it is when you have a vector multiplied by a given matrix that produces another vector, which can be represented as a scalar (eigenvalue) and the original vector; Found by solving the characteristic polynomial (they are the roots)
bisector
a point, ray, line, line segment, or plane that intersects the segment at its midpoint
Decagon
a polygon with 10 sides and vertices
Undecagon
a polygon with 11 sides and vertices
Dodecagon
a polygon with 12 sides and vertices
Triangle
a polygon with 3 sides and vertices
Quadrilateral
a polygon with 4 sides and vertices
Pentagon
a polygon with 5 sides and vertices
Hexagon
a polygon with 6 sides and vertices
Heptagon
a polygon with 7 sides and vertices
Octagon
a polygon with 8 sides and vertices
Nonagon
a polygon with 9 sides and vertices
Concave Polygon
a polygon with an indentation
reasonable conclusion
a position or opinion or judgment reached after considering facts and observations
Trapezoid
a quadrilateral that has exactly two parallel sides
Parallelogram
a quadrilateral that has two pairs of parallel sides
parallelogram
a quadrilateral whose opposite sides are both parallel and equal in length
unit rate
a rate that has a denominator of 1
rate
a ratio that compares two quantities measured in different units
percent
a ratio whose denominator is 100
matrix
a rectangular array of elements or entries set out by rows and columns
Matrix
a rectangular array of numbers inside square brackets
function
a relation that assigns exactly one output value for each input value
Square
a rhombus with four right angles
interval
a set containing all points or all real numbers between two given endpoints
orthogonal vector
a set of non-zero vectors; angle between vectors is 90; the dot product is zero
relations
a set of ordered pairs
Orthogonal Matrix
a square invertible matrix such that U-1=UT; has orthonormal columns and rows
cylinders
a three dimensional figure with two parallell, congruent circular basis connected by a curved lateral surface
Equianglular Triangle
a triangle in which all the angles are the same measure
Equilateral Triangle
a triangle in which the length of all the sides are equal
Isoceles Triangle
a triangle that has at least two sides of equal length
Obtuse Triangle
a triangle with an angle that is greater than 90 degrees
When multiplying two terms with the same base, you should ____ the exponents.
add
Obtuse Angle
an angle larger than a right angle
Acute Angle
an angle smaller than a right angle
estimate
an approximate calculation of quantity or degree or worth
symmetry
an attribute of a shape or relation
Function Notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x
linear equation
an equation whose answers are ordered pairs (x,y) and whose answers graph a straight line
Linear Equation
an equation whose graph is a straight line
Linear Inequality
an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line
linear inequality in 1variable
an inequality that can be written in the form ax+b<c where a,b,c are real numbers and a is not 0
Coordinate axes
lines that have the same scale and are drawn perpendicular to eachother.
Log Power Rule
logαMⁿ =nlogαM
Relative Minimum
lowest point on a graph
composed of
made up of
Synthetic Division
method used to divide polynomials by monomials
(a-b)(a-b) OR (a - b)²
a² -2ab + b²
(a+b)(a+b) OR (a+b)²
a²+2ab+b²
The system Ax-b is not homogeneous if...
b!=0
The system Ax=b is called homogeneous if...
b=0
polygon
closed plane figure having, literally, many angles and therefore many sides
best represents
closest to; most similar to
adding polynomials
combine all like terms
numerical expression
consists of numbers and operations
Natural Numbers
counting numbers
To compute the i-entry of Ax...
multiply every entry in the i-row of A by corresponding entries in x, take the sum of these products
.25 + 0 = .25
n + 0 = n
2/5 + ¼ = ¼ + 2/5
n + m = m + n
5 + 4 = 4 + 5
n + m = m + n
6 + 1.3 = 1.3 + .6
n + m = m + n
34 - 34 = 0
n - n = 0
5 - 5 = 0
n - n = 0
9.4 - 9.4 = 0
n - n = 0
¼ - ¼ = 0
n - n = 0
(.2)(0) = 0
n ∙ 0 = 0
(¾)(0) = 0
n ∙ 0 = 0
0 ∙ 1 = 0
n ∙ 0 = 0
0 ∙ 3 = 0
n ∙ 0 = 0
5 ∙ 0 = 0
n ∙ 0 = 0
(.6)(1) = 1
n ∙ 1 = n
(1)(1) = 1
n ∙ 1 = n
7/8 ∙ 1 = 7/8
n ∙ 1 = n
9 ∙ 1 = 9
n ∙ 1 = n
3 ∙ 1/3 = 1
n ∙ 1/n = 1
7 ∙ 1/7 = 1
n ∙ 1/n = 1
7/8 ∙ 8/7 = 1
n ∙ 1/n = 1
½ ∙ 2 = 1
n ∙ 1/n = 1
(-6)(-6) = (6)2
n ∙ n = n[sq]
coefficient
number in front of a variable
annual
occurring or payable every year
segments
set of points on a line that consist of two points called endpoints, and all the points between them
intersection
the act of intersecting as joining by causing your path to intersect your target's path
Right Angles
the angles made by perpendicular lines
region
the approximate amount of something usually used prepositionally as in 'in the region of'
Feasible Region
the area between intersecting lines (shaded area)
Leading Coefficient
the coefficient of the term with the highest degree
Span
the collection of all vectors in R^n that can be written as c1v1 + c2v2 + ... (where c1, c2, etc. are constants)
range
the difference between the highest and lowest scores in a distribution
absolute value
the distance a number is from 0 on the number line, value of n is written l n l
diameter
the distance across a circle through its center (is also twice the radius)
Diameter
the distance from one side of a circle through the center to the other side
Radius
the distance from the center of the circle to the outside edge
Symbol of Equality
the equal sign which indicates that two quantities are equal
Slope Intersept Form
the equation of a line in the form of y=mx+b, where m is the slope and b is the y intersept
T is 1-1 if and only if...
the homogeneous equation Ax=0 has only the trivial solution, has no free variables, has a pivot in every column, the n column of A are linear independent
Denominator
the number on the bottom of a fraction
Numerator
the number on the top of a fraction
Value
the number represented by a numeral
center
the point in the exact middle of a circle
Vertex
the point of intersection between lines
minimum
the point on a curve where the tangent changes from negative on the left to positive on the right
maximum
the point on a curve where the tangent changes from positive on the left to negative on the right
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
R is...
the set of all real numbers
Eigenspace
the set of all solutions for Ax=λx for a specific λ; the null-space of (A-λI)
domain
the set of all the input (x-values) for a function
range
the set of all the output (y-values) for a function
domain
the set of values of the independent variable for which a function is defined
perimeter
the size of something as given by the distance around it
slope of a line
the slope(m) of a line containing points (x(1), y(1)) & (y(1), y(2)) is given by: m = rise/run = y(2)-y(1)/x(2)-x(1) as long as x(2) does not equal x(1)
least common multiple
the smallest multiple that two or more numbers have in common
slope
the steepness of a line on a graph, equal to its vertical change divided by its horizontal change
slope
the steepness of a line on a graph, rise over run
Slope
the steepness of a line, equal to the ratio of a vertical change to the corresponding horizontal change
The set of all possible linear combinations of v_1, ..., v_p is called...
the subset of R^n spanned by v_1, ..., v_p, denoted by Span(v_1,...,v_p)
Mixed Number
the sum of a whole number and a fraction
v_1,v_2,...,v_p are independent if and only if...
the system has no free variables
The operation of reading any vector vER^n to p + v is called...
the translation by p
dependent
the variable in the relation whos value depends on the value of the independent variable
Independent Variable
the variable that makes up the domain
y-intercept
the y-coordinate of the point where the line crosses the y-axis.
output
the y-value in a function
Elimination Method
to solve a system of equations by adding equations to get rid of one variable. sometimes needing to multiply one equation by a # to make terms opposites
To find the parametric vector form for the solution set...
1) Row reduce the augmented matrix 2)Get a parametric description of the solution set 3) Write your solution x as a column vector; single out the coefficient for each variable in a column
If the system is consistent, these statements are true:
1) The equation Ax=b is always consistent for all bER^m 2) The column a_1, a_2,...,a_n fo A span R^m 3) the coefficient matrix has a pivot in every row
How to solve a linear system
1) Write down the augmented matrix B of the system 2) Row reduce B to Echelon form; if the last column of B is pivot, the system is inconsistent; if the last column of B is not pivot, it is consistent 3) Complete the row reduction to C = rref(B) 4) Write down the new system corresponding to C; locate basic/free variables 5) Use the new system to express all variables in terms of free variables
factor by grouping
1) arrange terms so the first two terms have a common factor & the last two have a common factor 2) for each pair of terms, factor out the the pair's GCF 3) if there is now a common binomial factor, factor it out 4) if no common binomial factor, begin again, rearranging terms differently. if no rearrangement works it can't be factored.
solving linear equalities in one variable
1) clear the equality of fractions by mult. both sides of ineq. by LCD of all fractions in the equality. 2) remove grouping symbols such as ( ) by using distributive property 3) simplify each side of equality by combining like terms 4) write the equality w variable terms on one side and numbers on the other by using addition prop of equalities 5) get the variable alone by using multiplication prop of equalities
solving linear inequalities in one variable
1) clear the inequality of fractions by mult. both sides of ineq. by LCD of all fractions in the inequality. 2) remove grouping symbols such as ( ) by using distributive property 3) simplify each side of inequality by combining like terms 4) write the inequality w variable terms on one side and numbers on the other by using addition prop of inequalities 5) get the variable alone by using multiplication prop of inequalities
multiplication property of inequality
1) if a,b,c are real numbers, and c is positive, then a<b & ac<bc are equivalent inequalities 2) if a,b,c are real numbers, and c is negative, then a<b & ac>bc are equivalent numbers. *the direction of the inequality symbol must be reversed for the inequalities to remain equivalent
Gauss elimination method
1) keep the x_1 term in one equation, eliminate x_1 from all others 2) keep the x_2 term in the second equation, eliminate x_2 from all next ones 3) and so on
Any linear system can have either...
1) no solutions 2) exactly 1 solution 3) infinite many solutions
problem-solving linear equations
1) understand the problem. read and re-read it. choose 2 variables to represent the two unknowns. construct a drawing if possible. propose a solution and check. 2) translate the problem into two equations 3)solve the system of equations 4) interpret the results: CHECK the proposed solution in the stated problem and state your conclusion
linear equation in 3 variables
1) write ea. equation in standard form: Ax+By+Cz=D 2)choose a pr of equations& use the equations to eliminate a variable 3) choose any other pr of equations & eliminate the same variable as in step 2 4) two equations & two variables should be obtained by step 2 & step 3. solve this system for both variables 5) to solve for the 3rd variable, substitute the values of the variables found in step 4 into any of the original equations containing the third variable 6) check the ordered triple solution in all three orig. equations
Relation:
A relationship between sets of infomation
to graph x<or=3
shade the numbers to the left of 3 and place a bracket at 3 on the number line. the bracket indicates that 3 is a solution:3 is less than or =to 3. in interval notation we write (-infinity,3] *may be easier to graph the inequality first then write it in interval notation. to help, think of the number line as approaching -infinity to the left or +infinity to the right. then simply write the interval notation by following your shading from left to right
finding slope given 2 points on a line
slope= change in y (vertical change)/change in x (horizontal change) to do this, choose two points of a line. label the two x-coordinates of two points x(1), x(2) [read "x sub one" and "x sub 2], and label the corresponding y-coordinates y(1) and y(2) the vertical change(rise) btwn these points is the diff in the y-coordinates: y(2)-y(1). the horizontal change(run) btwn the points is the diff of the x-coordinates: x(2)-x(1). the slope of the line is the ratio of y(2)-y(1) to x(2)-x(1), and we use the letter m to denote slope in this formula: m=y(2)-y(1)/x(2)-x(1)
The vector equation x_1a_1+x_2a_2+...+x_na_n = b and the system with the augmented matrix [a_1,a_2,...a_n|b] are equivalent if and only if
the corresponding system is consistent
T is onto if and only if...
the equation Ax=b is always consistent, the matrix A has a pivot in every row, the column of A spans R^m
The product Ax = x_1a_1+x_2a_2+...+x_na_n and is only defined when...
the number of columns in A equals the number of entries in x
multiplying the sum & diff of two terms
the product of the sum & difference of two terms is the square of the first term minus the square of the second term (a+b) (a-b) = a(sq)-b(sq)
R^n is...
the set of all vectors in n entries with real coefficients
slope-intercept form
y=mx+b
Commutative Property of Multiplication
(x)(2) = 2x
Factor the polynomial completely: 3x³ + 6x² + x + 2
(3x²+1)(x+2)
y- coordinate
(Ordinate) 2nd number in the pair.
*C = np*
*Cost* = number of items × price per item
Lesson *1-9*
*Opposites and Absolute Values*
*P = 2l + 2w*
*Perimeter of rectangle* = ( 2 × length ) + ( 2 × width )
Step 3
*Reread the problem and* *write an equation.*
Plan for Solving a Word Problem *Step 4*
*Solve the equation and* *find the unknowns* *asked for.*
Lesson *1-6*
*Translating Problems into Equations*
Lesson *1-5*
*Translating Sentences into Equations*
Lesson *1-4*
*Translating Words into Symbols*
Chapter 1 Section 1
*Variables and Equations*
Lesson *1-1*
*Variables*
*1. If a is positive,*
*then −a is negative.*
*2. If a is negative,*
*then −a is positive.*
3. If a is zero,
*|a| = 0.*
1. If a is positive,
*|a| = a.*
2. If a is negative,
*|a| = −a.*
Subspaces
1. The zero vector is in H; 2. For u and v in H, u + v is also in H; 3. For u in H, cu is also in H (c is a constant)
What will the solution of this absolute value equation look like |exp.| < a ?
-a < exp < a
combinations
...
LU Factorization
1. Ly = b; Ux = y; 2. Reduce A to echelon form; 3. Place values in L that, by the same steps, would reduce it to I
Multiplication Property of Zero
0x = 0
What are some Complex Numbers?
1 + 2i, -3i, 4, 3 - 11i, etc. Every Real Number is a Complex Number (Ex. 4 --> 4+0i).
x⁻⁵
1 / x⁵
x⁻⁵/⁶
1 / ⁶√x⁵
Matrix multiplication warnings
1. AB != BA ; 2. If AB = AC, B does not necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0
Echelon form
1. All nonzero rows are above any all zero rows; 2. Each leading entry is in a column to the right of the previous leading entry; 3. All entries below a leading entry in its column are zeros
When there are no grouping symbols, simplify in the following order:
1. Do all multiplications and divisions in order from left to right. 2. Do all additions and subtractions in order from left to right.
Invertibility rules
1. If A is invertible, (A^-1)^-1 = A; 2. (AB)^-1 = B^-1 * A^-1; 3. (A^T)^-1 = (A^-1)^T
Identity Property of Multiplication
1x = x
A linear system where two lines are directly on top of each other (infinite answers) would be what type of system?
A dependent system.
*grouping symbol*
A device used to enclose an expression that should be simplified before other operations are performed. Examples: *parentheses, brackets, fraction bar.*
fraction
A division problem represent as such: 3/4.
*open sentence*
A sentence containing one or more variables.
orthogonal set
A set of vectors where Ui . Uj = 0 (and i != j); if S is an orthogonal set, S is linearly independent and a basis of the subspace spanned by S
heptagon
A seven-sided polygon.
Synthetic Division...
A shortcut that can be used to divide h-k into a polynomial. Make sure that everything is written in decreasing order of powers!
hexagon
A six-sided polygon.
prime number
A whole number that has exactly two factors, 1 and itself.
Formula
An algebraic equation that shows the relationship among specific quantities.
obtuse angle
An angle that is larger than a right angle but less than a straight angle.
acute angle
An angle that is smaller than a right angle.
octagon
An eight-sided polygon
undeagon
An eleven-sided polygon.
*formula*
An equation that states a rule about a relationship.
formula
An equation that shows a relationship among two or more quantities
*numeral*
An expression that names a particular number; a *numerical expression.*
Algebraic Expression
An expression that contains numbers, operations and variables.
Associative Property
Change the grouping without changing the outcome.
Disjoint Sets
Empty Sets { } or null.
Verticies
Endpoints of the sides.
Union
Everything that belongs to BOTH sets.
When dealing with shifting of graphs (ex.)...
Ex. from Exam 2... 1. f(x) = x^2 + 2x - 7; left 8 unit, up 11 units. 2. f(x+8) + 11 3. f(x+8) = (x+8)^2 + 2(x+8) - 7 + 11. Clean up and done!
Combining Transformations...
Ex. y = -2(x - 1)^2 + 3 - Reflects across the x-axis. 2 Stretches vertically by a factor of 2. - 1 Shifts to the right 1 unit. + 3 Shifts upward 3 units.
f(2x):
Example of a horizontal shrink
f(½x):
Example of a horizontal stretch
- f(x):
Example of a reflection about the X axis
f(-x) :
Example of a reflection about the y axis
½ f(x) :
Example of a vertical shrink
2f(x):
Example of a vertical stretch
What is a Polynomial Function?
For example, 2x^2 - 3x + 1 is a Polynomial; C(X) = 4 is a Polynomial... you just don't see the "0's."
signed numbers
Numbers designated as either negative or positive by prefixing the number with either a (-) or a (+).
f(x) = ±√ X :
Not a function because it would have two answers; + 9 or - 9
non-included
Not between
Example: a/0
Not defined.
Base
Number getting multiplied by itself.
Coefficient
The 2 in the term 2x
Constant
The 3 in the expression 2x + 3
What is the Domain of a Function?
The Domain of a Function is the set of values that makes the function well defined. If the function is not well defined at a point, then that point is not in the domain.
With Verticle Lines...
The Slope is undefined!
These two equations would be easiest to solve by what method Equation¹: 4x + 2y = 6 Equation²: 3x - 2y = 16?
The addition method. If set up as if adding them together the y's would cancel. Then you could solve for x and then solve for y.
right angle
The angle made by perpendicular lines.
Quotient
The answer to a division problem.
Product
The answer to a multiplication problem.
Difference
The answer to a subtraction problem.
Sum
The answer to an addition problem.
*area*
The area of a region is the number of square units it contains.
What is a Parabola?
The graph f(x) is called a Parabola. Examples: f(x) = 2x^2 - x + 1 (Opens Upward because positive. a>0.) g(x) = -x^2 + 3x -5 (Opens Downward because negitive. a<0.) The highest (Absolute Max)/lowest (Absolute Min) point is called the Vertex.
Greatest Common Factor
The greatest factor that is common to two or more numbers.
perimeter
The measure (or distance) around a polygon.
What is Median?
The median (middle) value of a range of values.
*coordinate* *of a point*
The number paired with that point on a number line.
*value* *of* *a* *numerical* *expression*
The number named by the expression.
Coefficient
The number mulltiplied by a variable in a term.
Exponent
The number of times a factor in repeated in a power.
Dimension
The number of vectors in any basis of H; the zero subspace's dimension is 0
denominator
The number that is on the bottom in a fraction.
numerator
The number that is on top in a fraction.
commutative property for addition
The order in which two real numbers are added does not affect the sum.
*graph* *of a number*
The point on a number line that is paired with the number.
Vertex
The point in a function where the function reaches a min or a max
Unit Conversion
The process of renaming a measurement using different units.
multiplication identity property
The product of any number and one is that number.
mixed number
The sum of a whole number and a fraction.
widest
The widest/narrowest part of the nuchal translucency should be measured.
x-intercept
The x-coordinate of the point where a line crosses the x-axis.
Range
The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs.
*origin*
The zero point on a number line. The intersection of the axes on a coordinate plane.
Write the equations for two lines that are parallel.
There are many answers. An example would by y = 2x and y = 2x + 7.
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 types of Transformations are there?
There are many types of Transformations. All moving either left, right, up or down. Some are... y = f(x) + c --> Upward. y = f(x) - c --> Downward. y = f(x - c) --> Right. y = f(x + c) --> Left. If a number is in front of the X, this number makes it go slower (Ex. 1/2) or faster (Ex. 2).
What is an Imaginary Unit?
There is also more, but the most commonly used is the one above and i^2 = -1.
What is true about any horizontal line and a vertical line?
They are perpendicular
domain in interval notation f(x)=⁴√x-5 ( for an even index, root must be positive)
[5,∞)
greater than or equal to x, less than or equal to y
[x, y]
Parabola
a "u" shaped graph y=x²
Multiplication Property of Zero
a * 0 = 0
Inverse Property of Addition
a + (-a) = 0
Identity Property of Addition
a + 0 = a
Commutative Property of Addition
a + b = b + a
³√ a = :
a 1/3
scale model
a model of an object in which the dimensions are in proportion to the actual dimensions of the object.
grid
a network of horizontal and vertical lines that provide coordinates for locating points on an image
Eigenvector
a nonzero vector (x) where Ax=λx;basically its a scaled vector, where the linear transformation doesn't change; It's also a nonzero in the nullspace of a given matrix [think(A-I)x=0
Complex Number
a number in the form a+bi
constant
a number representing a quantity assumed to have a fixed value in a specified mathematical context
Zero
a number that can be used to describe a physical distance of no magnitude
irrational number
a number that can not be written a/b
perfect square
a number that has a whole number for a square root ex: 9, 16, 121
composite
a number that has more than two factors
coordinate
a number that identifies a position relative to an axis
Irrational Number
a number that would take an infinite number of digits to express
Dividend
a number to be divided by another number
Decimal
a number with one or more digits to the right of the decimal point
prime factorization
a number written as the product of its prime factors
Absolute Value
a number's distance from zero
Monomial
a number, variable, or a product of a number and variable
rectangle
a parallelogram with four right angles
additional
existing or coming by way of addition
What will the solution of this absolute value equation look like |exp.| = a ?
exp = a or exp = -a
satisfy
fill or meet a want or need
solving an equation
finding all the solutions of an equation
systems
groups of organs working together to perform complex functions
Determine the possible number of positive real zeros, negative real zeros, and imaginary zeros for the function: f(x) = x⁴ + 3x³ - 2x² - x + 10
positive zeros: 0 or 2, negative zeros: 0 or 2, imaginary zeros 0, 2, or 4
counterexample
refutation by example
Polygon
simple, closed, flat geometric figures whose sides are line segments and whose lines do not cross
multiple
skip-counting by any given number Ex: 4, 8, 12, 16...
closets to
something that is close to another object
cube
special polyhedron with faces that are all squares
A system of linear equations (a linear system) is...
x_1,x_2,...,x_n a collection of linear equations that is the value of the same set of variables
Using long division: (x⁴ + 10x³ + 8x² - 59x +40) ÷ (x² + 3x -5)
x² + 7x - 8
Perform the indicated operation: (x - 2)(x+3)(x-5)
x³ - 4x² - 11x + 30
Perform the indicated operation: (2x⁴+9x-7) - (x⁴+6x+5)
x⁴ + 3x - 12
Point slope form:
y - y1 = M ( X - X1 )
The point slope form of a linear equation is ________
y - y1 = m(x − x1)
What is the point slope equation? Used when you have a slope and a random point.
y - y¹ = m(x - x¹)
⁴√7 * ⁴√2
⁴√14
Any row/column is...
a row/column with 1 nonzero entry in it
A matrix with one column is called a...
column vector
The homogeneous system is always...
consistent, x=0 is always a solution
Any given matrix A is row equivalent to...
exactly one matrix B in Reduced Echelon form; B = rref(A)
addition property of inequality
if a, b, and c are real numbers, then a<b & a+c<b+c are equivalent inequalities
quotient rule for exponents
if m & n are positive integers & a is a real number, then a(m)/a(n)=a(m-n) as long as a is not 0
power of a product rule
if n is a positive integer and a & b are real numbers, then (ab)(n) = a(n)b(n)
Augmented matrix transformation is...
linear
A matrix is...
a rectangular array of numbers aligned in rows and columns
Verticle line test:
Given the graph of a relation, if you can draw a verticle line that crosses the graph in more than one place, the relation is not a function
The identity matrix is...
I_n, nxn matrix
The arguement of a function:
Is the " X " in f(x)
Symmetric about the Y-axis:
Whatever the graph is doing on one side of the Y-axis is mirrored on the other side
