Math Final Review 2019
Give an example of an algebraic equation that represents a function.
y = 100 + 20x
How do we interpret direct proportion?
If y/x = k or y = kx, where k is a constant value, then y is said to be directly proportional to x. The constant value k in a direct proportion is called the constant of proportionality. The graph of a direct proportion is always a line through the origin but does not lie along the horizontal or vertical axis. The constant of proportionality in a direct proportion often represents a unit rate k. In general, you can use the point (1,y) on a direct proportion graph to find a constant of proportionality. You can then use the unit rate to write a direct proportion equation y = kx.
Explore and apply the properties of vertical angles.
In the diagram, <a and <b are vertical angles. Because <a and the hypothetical <c are adjacent angles that form a straight line, they are supplementary, as is <b and the hypothetical <d. It can be deduced that m<a + m<c = 180 degrees, and m<b + m<d = 180 degrees.
Explain the Substitution Method.
In the substitution method, we find the solution by rewriting one equation and then substituting it into the other equation. EX: 4x + y = 7 (1) 3x + 2y = 9 (2) Rewrite (1) as y = -4x + 7, and substitute it into (2): 3x + 2(-4x + 7) = 9 Now we can solve for x: 3x - 8x + 14 = 9 -5x = -5 x = 1 Substituting x = 1 into either of the equations show that y = 3, so the solution is (1,3).
Explain the formula for the volume of a rectangular prism.
In this example, pretend that there is a rectangular prism that has a length of 5 ft., a width of 6 ft., and a height of 2 ft. The formula for rectangular prisms is V = lwh, so we plug in the respective units. V = lwh V = 5 x 6 x 2 V = 60 ft^3 The volume of the prism is 60 cubic feet. Note that the same formula is used for cubes.
What are integers, whole numbers, and natural numbers?
Sub-categories for rational numbers. Integers: All whole numbers and their negative counterpoints(EX: ... -4, -3, -2, -1, 0, 1, 2, 3, 4...). Whole Numbers: A number with no fractional or decimal part. Cannot be negative. EX: 0, 1, 2, 4... Natural Numbers: Whole Numbers from 1 and up(EX: 1, 2, 3, 4, 5...).
Explain the Graphical Method.
The Graphical Method is when we graph each linear equation and finding where they intersect. EX: x + y = 5 2x - y = 4 Rewrite the equations into slope-intercept form. y = -x + 5 y = 2x - 4 Graph the equations by using the y-intercept and slope for each. If there were a graph here, we would see that the two lines intersect at (3,2). The solution to this pair of linear equations is (3,2).
What is absolute value?
The absolute value of a number isn't its distance from zero on the number line. It is thus always positive. We indicate absolute value by putting two bars around the number. Example: |-4|(the absolute value of -4) is 4, because it is four spaces away from zero on the number line.
Angle
The figure formed when two rays share a common endpoint(vertex). Symbol: <ABC/<CBA/<B When naming an angle with three letters, the middle letter should be the vertex. If an angle is named by the vertex letter, be sure the vertex is not shared by another angle.
Define the parts of an exponent and show which is which on this exponent: 10^4
The power is the number of times the base is multiplied by itself(10 X 10 X 10 X 10). The base is the big number(10). The exponent is the small, raised number(4).
Writing a Two-Step Equation
The sum of 400 and 15 times a number is 700. Let n represent the number 400 + 15n = 700
Give an example of a verbal description of a function.
The total amount Janice pays equals $100 registration fee plus $20 times the number of hours she takes lessons.
What is volume?
The volume of a 3-D figure refers to the number of cubic units needed to fill the figure. Or, put more simply, "How much will fit in here?" The answer is volume.
Standard Notation
The way we usually write numbers. EX: 2,300,000
Supplementary Angles
Two angles whose sum is 180 degrees. They do not need to be adjacent. Symbol: A straight angle with an acute angle at the right end.
Complementary Angles
Two angles whose sum is 90 degrees. They do not need to be adjacent. Symbol: Two angles whose degrees add up to 90.
Perpendicular Lines
Two lines that meet and form right angles(90 degrees). Symbol: An upside-down 'T'
How does slope and unit rate relate to each other?
Unit rate, or a ratio that shows the units a line rises per 1 line it runs, is another way to understand slope, as slope represents unit rate. EX: Slope = Rise/Run = 2/3 /1 This says: For every time a line runs run, it rises 2/3.
Explain how we can use rate of change to tell if a function is linear.
We can tell if a function is linear by finding rate of change and determining if it is constant. If it's constant, the function is linear; if not, it's nonlinear.
Quotient of Powers Rule
When dividing powers of the same base, keep the base and subtract the exponents. EX: 5^9/5^7 = 5^9 - 7 = 5^2
How does combing like terms work?
When we combine like terms, we rewrite the expression so that it contains fewer numbers, variables and operations. EX: a + a + a + a + a + a can be simplified to 6a. The 6 is the coefficient, which tells us how a's we have. Terms can only be combined if they are like terms. They can have different coefficients, but they have to have the same variables raised to the same powers. EX: 6a + a + 7a can be simplified to 14a.
What is cube root?
When we cube an number, we raise it to the power of 3. The opposite is a number's cube root. It is indicated by 3√. EX: 3√8 = 2, 3√27 = 3, 3√1/125 = 1/5. When finding a cube root, ask yourself, "What number to the third power equals the number in the √?"
What is square root?
When we square a number, we raise it to the power of two. The opposite is a number's square root. The square root of a number is indicated by putting it in a √. Example: √16 equals 4. Ask yourself, "What number times itself equals the number in the √?"
Power to a Power Rule
When you have an exponent raised to another exponent, you keep the base and multiply the exponents. EX: (4^2)^3 = 4^2 X 3 = 4^6
Name the rules for solving an equation.
1. Combine like terms or distribute (if possible). 2. Undo the addition or subtraction using the inverse operation. 3. Undo the multiplication or division using the inverse operation. 4. Check. EX: x + 7 = 13 x + (7 - 7) = (13 - 7) x = 6 Check: x + 7 = 13 6 + 7 = 13 13 = 13
Show how to use these steps in an equation: 1. Use Distributive Property 2. Combine Like Terms 3. Use Inverse Operations
3x + 2(2x - 1) = 33 3x + 4x - 2 = 33 7x - 2 = 33 7x + (-2 + 2) = (33 + 2) 7x/7 = 35/7 x = 5
HELPFUL SHORTCUT FOR ROTATIONS (DO NOT FORGET!!!!!!!!!!!!!!!!!!!!!!!!!)
90 degrees clockwise/270 degrees counterclockwise: (x,y) > (y,-x) 90 degrees counterclockwise/270 degrees clockwise: (x,y) > (-y,x) 180 degrees clockwise/180 degrees counterclockwise: (x,y) > (-x,-y)
What is a cluster and a outlier?
A cluster is a set of closely grouped data, and an outlier is a data point that is very different from the rest of the data in the set.
What is a coordinate plane?
A coordinate plane is a flat surface formed by the intersection of two lines or axes: the horizontal line, known as the x-axis, and the vertical line, known as the y-axis. The x- and y-axes intersect(cross) at the origin. The coordinate plane is divided into four quadrants. In quadrant I, both values are positive, in quadrant II, the x-values are negative, while the y-values are positive, in quadrant III, both values are negative, and in quadrant IV, the x-values are positive, while the y-values are negative.
What is dilation?
A dilation is a transformation that enlarges or reduces a figure by a scale factor. The scale factor is the amount by which you stretch or shrink the original figure. When you change the size of a figure, there is a center of dilation, which is a fixed point in the coordinate plane around which the figure expands or contract
Explain what a function is.
A function assigns exactly one output to each input. Your x-values will not repeat in a function. All functions are relations but not all relations are functions.
Show a mapping diagram
A mapping diagram uses arrows to map each input onto one or more outputs.
Negative Exponent Property
A negative exponent in the numerator becomes a positive exponent when moved to the denominator(x^-m = 1/x^m). A negative exponent in the denominator becomes a positive exponent when moved to the numerator(1/x^-m = x^m).
What is a prime number?
A number that has only 1 and itself as factors. Includes 2, 3, 7, and 13, amongst others. The multiples of 2, 3, and 5, besides the numbers themselves, are composite, meaning they have more than two factors.
Ray
A part of a line that has one endpoint and extends forever in one direction. Symbol: A straight arrow going right above the two points. The endpoint is named first.
Line Segment
A piece of a line with two endpoints. Symbol: A straight line above the points.
What is a reflection?
A reflection is a transformation that flips a figure over a line of symmetry. In the chart, the original figure and the image are congruent. They are the same distance away from the line of symmetry, so we can say that they are equidistant from the line of symmetry. To do a reflection, move each point according to the given criteria.
Explain what a relation is
A relation is a set of ordered pairs(like the x- and y-coordinates are in a "relationship"). In a relation, all of the x-coordinates are called the domain, and all of the y-coordinates are called the range. EX: Name the domain and range for this relation: (-5,-3)(-2,0)(1,3)(4,6)(7,9) Domain: {-5, -2, 1, 4, 7} Range: {-3, 0, 3, 6, 9} Always list the domain and range in numerical order.
Inverse Operation Rules
Addition = Subtraction Subtraction = Addition Multiplication = Division Division = Multiplication Squaring = Square Root Cubing = Cube Root
Solving 1-Step Equations
Addition: y + 14 = 20 - 14 - 14 ----------- y = 6 Subtraction: x - 120 = 80 + 120 + 120 ------------- x = 200 Multiplication: 3n = 12 3n/3 = 12/3 n = 4 Division: k/2 = 16 k/2 x 2 = 16 x 2 k = 32
What is a real number?
All numbers that can be found on a number line. These include all rational and irrational numbers. Irrational means it can't be written as a simple fraction. √2 is an example, as is 3.14159265...
Acute Angle
An angle measuring less than 90 degrees and more than 0 degrees.
Obtuse Angle
An angle measuring more than 90 degrees and less than 180 degrees.
Terms for the Number of Solutions to an Equation
An equation with one solution is called consistent. EX: 3(x - 4) = 2(x - 1) An equation with an infinite amount of solutions is an identity. EX: 3x + 5 = x + 2x + 5 An equation with no solution is called inconsistent. EX: 5(x + 3) = 5x + 3
Point
An exact location in space. Symbol: The name of the point (say it like this: 'point (letter)'.
What is an ordered pair?
An ordered pair gives the coordinates of a point. They are called an "ordered pair" because the order matters. The x-coordinate always comes first, then the y-coordinate, like so: (x,y).
Corresponding Angles
Angles that are in the same position on two parallel lines in relation to a transversal line. Symbol: Four pairs. Corresponding angles are congruent to each other.
What is a rational number?
Any number that can be written as a fraction or ratio. EX: 1/2( = 0.5), 0.25( = 1/4), -7( = -7/1), 4.12( = 412/100). If a decimal has a line over a number, than the line means it repeats forever(0.333333333...). Ones that don't stop eventually and are called terminating decimals.
Zero Exponent Rule
Anything raised to the zero power = 1 EX: 6^0 = 1
Represent a relationship between two variables using a linear equation.
B: 1 2 3 4 5 D: 4 5 6 7 8 D is always 3 more than B. You can represent the relationship between their ages using a linear equation with two variables. This can be written as D = B + 3.
Solve for a variable in a two-variable linear equation.
Consider the formula for the perimeter, P units, of a square with a side length of e units, P = 4e. You can use this formula to find the value of e when you know the value of P. For example, if P = 18, you can find the value of e by substituting the value of P into the equation and solving for e. P = 4e 18 = 4e 18/4 = 4e/4 4.5 = e Suppose you are given many values of P and asked to find the corresponding values of e. You may find it convenient to solve the equation for e. That is, you can express e in terms of P before substituting values of P. P = 4e P/4 = 4e/4 P/4 = e 18/4 = e 4.5 = e
Conversion of a Positive # from Sta. Not. to Sci. Not.
Count how many places you have to move so that there is only a number in between 1 and 10 that remains. The number of spaces you move the decimal point is related to the exponent of 10. If it's greater than 1, it's positive, and if it's less than 1, it's negative. EX1: 3,320,000. Move the decimal point six spaces to the left to get 3.32. The exponent of 10 will be positive 6. EX2: 0.0007274. Move the decimal point four spaces to get 7.274. The exponent of 10 will be negative 4.
Give an example of the different ways to write an exponent.
Exponential form: 10^4 Word form: Ten to the fourth power Standard form: 10,000 Expanded form: 10 X 10 X 10 X 10
Alternate Exterior Angles
Exterior angles that lie on opposite sides of the transversal. Symbol: Two pairs. Alternate exterior angles are congruent to each other.
Solving Two-Step Equations
First, add/subtract both sides, then multiply/divide both sides. EX: 3x + 7 = 28 3x + (7 - 7) = (28 - 7) 3x = 21 3x/3 = 21/3 x = 7
Power of a Product Rule
For expressions with the same exponent, distribute the exponent to each base((a X b)^m = a^m X b^m). To find the product of two algebraic expressions with the same exponent, multiply their bases(a^m X b^m = (a X b)^m).
Power of a Quotient Rule
For expressions with the same exponent, distribute the exponent to each base((a/b)^m = a^m/b^m). To find the quotient of two algebraic equations with the same exponent, divide the bases(a^m/b^m = (a/b)^m).
Explain the formula for the volume of a cylinder.
For this example, pretend that there is a cylinder with a radius of 3 ft. and a height of 5 ft. The formula is V = πr^2h. If the diameter is shown instead of the radius, multiply it by 1/2 to find the radius. V = πr^2h V = 3.14 x 3^2 x 5 V = 141.3 in^3 The volume of the cylinder is 141.3 cubic inches. Remember, when multiplying, substitute π for 3.14.
Explain the formula for the volume of a cone.
For this example, pretend that there is a radius of 6 in. and a height of 8 in. The formula is V = 1/3πr^2h. Remember that r = 1/2d. V = 1/3πr^2h V = 1/3 x (π x 6^2 x 8) V = 1/3 x (3.14 x 36 x 8) V = 301.44 in^3 The volume of the cone is 301.44 cubic inches. Remember, π = 3.14.
Explain the formula for the volume of a sphere.
For this example, pretend that there is a sphere with a radius of 6 in. The volume is V = 4/3πr^3. Remember, the formula for radius, if not shown, is r = 1/2d. V = 4/3πr^3 V = 4/3 x (3.14 x 6^3) V = 4/3 x (3.14 x 216) V = 904.32 in^3 The volume of the prism is 904.32 cubic inches. Remember, substitute π with 3.14 when multiplying.
Explain the formula for the volume of a triangular prism.
For this example, pretend that there is a triangular prism with a base of 4 ft. and a height of 18 ft., and the front and back have a height of 12 ft. The formula is V = 1/2bhl The h is the height of the front and back, and the l is the height of the prism itself, which is also the length. V = 1/2bhl V = 1/2 x (4 x 12 x 18) V = 432 ft^3 The volume of the prism is 432 cubic feet.
Give an example of a translation.
Given △ABC, translate it as follows: (x + 4, y + 3). First, you write the original coordinates: A: (-2,-2) B: (-2,-4) C: (-4,-3) Then, you calculate each translated point by adding 4 to he x-value and 3 to the y-value. Lastly, plot and label the image as A'B'C'. A': (2,1) B': (2,-1) C': (0,0) Reminder: If you translate a figure more than once, you can label the new figures with double prime(''), triple prime('''), or more. Remember to write the relation like this: △ABC ≅ △A'B'C'.
Give an example of a rotation.
Here, we will rotate △ABC 90 degrees clockwise. A: (-1,2) B: (-1,3) C: (-4,1) A': (2,1) B': (3,1) C': (1,4) The x- and y-coordinates swapped places and then took the appropriate signs for quadrant I.
Why does a y-intercept matter?
If an equation has a y-intercept, then the points it crosses will be different than if it doesn't.
Conversion from Sci. Not. to Sta. Not.
If the exponent is positive, move the decimal to the right. (EX: 8.91 x 10^7. The exponent (7) is positive, so move the decimal seven spaces to the right and fill with zeroes.) If the exponent is negative, move the decimal to the left. (EX: 4.667 X 10^-6. The exponent (-6) is negative, so move the decimal six spaces to the left and fill with zeroes.)
How can we determine if a function is linear through a graph?
If the slope is straight, the function is linear; if it's curved, the function is nonlinear.
Explain the Pythagorean Theorem.
A right triangle has two "legs" and a hypotenuse - the longest side of a right-angled triangle, which is always the side opposite the right angle. The two legs are connected at the right angle. a and b are the length of the legs(it doesn't mater which is which). The length of the hypotenuse, c, is always longer than the length of leg a or the length of leg b. The Pythagorean Theorem is used to find the length of a side of a right triangle. The formula is a^2 + b^2 = c^2. In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. In this example, a = 3 and b = 4. a^2 + b^2 = c^2 3^2 + 4^2 = c^2 9 + 16 = c^2 25 = c^2 To isolate c, find the square root of both sides. √25 = √c^2 c = 5 The length of the hypotenuse is 5 inches. The theorem can also be used to the missing length of a leg. Here, a = 6 and c = 10. a^2 + b^2 = c^2 6^2 + b^2 = 10^2 36 + b^2 = 100 -36 -36 b^2 = 64 √b^2 = √64 b = 8 The length of b is 8 centimeters.
What is a rotation?
A rotation is a transformation that turns a figure around a fixed point called the center of rotation. The number of degrees that the figure turns is called the angle of rotation. The criteria tells us the degrees that the shape moves, whether the shape moves clockwise or counterclockwise, and the center of rotation. A rotation does not change the size or shape of the figure. This means the image after the rotation is congruent to the original figure. So that means that ABC was rotated 90 degrees in a counterclockwise direction to form A'B'C Also, the two triangles are congruent--the corresponding sides are the same length and the corresponding angles are the same degrees. Rotations can also be performed on a coordinate plane. Usually, the origin (0,0) will be the center of rotation.
What is a scatter plot?
A scatter plot is a type of graph that shows the relationship between two sets of data. Scatter plots graph data as ordered pairs(this is simply a pair of numbers or mathematical objects--but the order in which they appear together matters. Example: After a test, Ms. Phinney asked her students how many hours they studied. She recorded their answers, along with their test scores: (Read like this: Name: # of hours studied, score) Albert: 0, 54 Bryan: 0.6, 64 Clara: 0, 69 Daniel: 1, 70 Eddy: 1, 75 Frank: 1.5, 80 Gabby: 1, 82 Harold: 2, 85 Isabella: 2, 93 Jack: 2.5, 95 Kyle: 3.5, 95 Lauren: 3.5, 99 Mark: 4, 100 The data is plotted onto the graph to see if there is a relationship between the number of hours studied and test scores. The relationships, called correlations, are positive(both sets increase), negative(one goes up and the othe goes down), and none(no relationship at all). The line drawn is the line of best fit, which roughly describes the relationship between the number of hours studied and test scores.
Line
A straight arrangement of points that extend indefinitely in opposite directions. Symbol: An arrow going in both directions above the points.
What is a system of linear equations?
A system of linear equations is when we take two linear equations and study them together. EX: ax + by = c and dx + ey = f Because each of the linear questions represents a line, we can ask, "If I draw two lines, where do they intersect?"
What is a term in an expression?
A term is a number by itself or the product of a number and at least one variable. EX: 5(number by itself), x(variable by itself), 7y(number and variable together), 16mn^2(number and more than one variable together) Terms are separated by an addition calculation, which may appear as a positive or negative sign.
What is a transformation?
A transformation is a change of position or size of a figure. When a figure is transformed, a new figure that is related to the original is created.
What is a translation?
A translation is a transformation that moves all the figure's points the same distance and direction. However, the orientation and size remain the same. The newly translated figure is called the image, and the new points are written with a prime symbol('). In the chart, the lines have the same length and the angles are the same number of degrees. They are simply plotted on different parts of the coordinate plane, so only the locations are different.
What is a two-way table?
A two-way table is a lot like a regular table, except that it show some two or more sets of data about the same subject. The data relates to two or more different categories or qualities. You use a two-way table to see if there is a relationship between the categories. EX: Mr. Nayeli collects data from the students in his class about whether they can or cannot bike to school. From the totals, we can see that there are 30 students, and we can answer questions like: -How many kids can bike to school? 16. -How many boys can't bike to school? 4. -How many girls can bike to school? 9. -How many students can't bike to school? 14. The common subject is the students, and we can conclude that if you are a boy, you are more likely to be able to ride a bike to school than a girl.
Scientific Notation
A way of writing numbers that are too big or too small to be conveniently written in decimal form. The first number is in between 1 and 10, and the second number is a power of 10. EX: 7.4 X 10^9 = 7,400,000,000(very large); 7.4 X 10^-9 = 0.0000000074
Give an example of a reflection.
In this example, we are reflecting △EFG over the x-axis. First, count how many units each point is away from the line or symmetry(in this case, the x-axis) and draw the reflected point the same distance away on the other side. It's easier to work point by point than to move the whole figure at once. Lastly, plot and label the image as E'F'G'. E: (1,-2) F: (4,-4) G: (2,-4) E': (1,2) F': (4,4) G': (2,4) Shortcut: When a figure is reflected across the x-axis, the sign of the y-value will simply change to the opposite, and when a figure is reflected across the y-axis, the sign of the x-value will simply change to the opposite.
Explain the Elimination Method.
In this way, we are able to "get rid of" one of the variables through addition or subtraction. Sometimes we also have to multiply one or both equations by a factor that will make it possible to eliminate one of the variables. Once we are left with one variable we find the value of it. Then we can use substitution to find the other variable. EX: 2x - y = 1 x + y = 2 Because the y's are opposite integers, hey cancel out. Then, we add the x's. 3x = 3 3x/3 = 3/3 x = 1 Now substitute x = 1 into either of the original equations to find y. 2(1) - y = 1 2 - y = 1 y = 1 Check the solution with the other equation. x + y = 2 1 + 1 = 2
Alternate Interior Angles
Interior angles that lie on opposite sides of the transversal. Symbol: Two pairs. Alternate interior angles are congruent to each other.
Parallel Lines
Lines that never intersect. Symbol: Two straight lines
Explain the four types of relation.
One to One: Each input is mapped onto exactly one output. One to Many: One input is mapped onto many outputs. Many to One: Many inputs mapped onto one output. Many to Many: One input is mapped onto many outputs, and one output is related to many outputs.
Vertical Angles
Pairs of angles formed by intersecting lines. Vertical angles are congruent(An = with a ~ above it.). Symbol: <A,<B
How can we compare two linear functions represented in the same form?
Rate of change is one way to compare, but another way is to compare with functions. EX: y = 15x y = 20x Equation B is bigger. Finally, a verbal description can also be used to compare functions.
Dividing Numbers in Sci. Not.
Remember the shortcut for dividing numbers with the same base and subtract the exponents, but keep the base. EX: 8 X 10^9/4 X 10^6 = 8/4 X 10^9/10^6 = 2 X 10^3
Multiplying Numbers in Sci. Not.
Remember the shortcut for multiplying numbers with the same base and add the exponents, but keep the base. EX:(2 X 10^4)(3 X 10^5) = 2 X 10^4 = 2 X 3 X 10^9 = 6 X 10^9
What if a variable has no coefficient?
Secretly, it has a coefficient of 1. EX: 'm' = '1m', 'k^3' = '1k^3'
Show an example of a graphed function.
See Big Fat Notebook, Page 418
What is a slope?
Slope is commonly referred to as the steepness of a line. More specifically, slope is a number that is a ratio that describes the tilt of a line: Slope = Rise/Run Rise is how much a line goes up or down. Run is how much a line moves left or right. EX: A slope of 2/3 means that every time the line rises 2, it also runs 3.
What are statistics and data?
Statistics is the study of data, and data is a collection of facts -- sometimes these facts come I the form of numbers, words, or descriptions. There are two types of data: Quantitative: information given in numbers. Usually this is information that you can count or measure. Qualitative: information given that describes something. Usually this is information that you can observe, such as appearances, textures, smells, tastes, etc.