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Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 1: HANDS ON EQUATIONS
Part 1: Solving 2-Step Equations
The goal of solving equations is to get the variable ___________________
To do this, you will need to first _________ or _______________, and then _____________ or ____________
Example: 4x – 2 = 10
Add 2 4x = 12
Divide by 4 x = 3
You Try!
1. 2x + 3 = 15 2. 9x – 1 = 8
Part 2: Solving Multi-Step Equations
Always start by ___________________________________ on either side of the ___________________
Example: 4x – x + 5 = 2x – 4 + 15
Combine like terms 3x + 5 = 2x + 11
Subtract 5 3x = 2x + 6
Subtract 2x x = 6
You Try!
1. 2x + 4x – 5 = 4x + 12 – 3 2. 5x + 2 – 9 = 3x + x – 7 + 3
Part 3: Equations with Negatives
When solving equations, be careful with the _______________________!
A negative in front of a positive, will be ________________
Example: -5(3x) = -15x
A negative in front of a negative, will be ________________
Example: -2(-4x) = 6x or 4 – (-3) = 4+3 = 7
EXAMPLE: -3(2x) + 5 = -x – 10
Multiply -3 -6x + 5 = -x – 10
Add x -5x + 5 = -10
Subtract 5 -5x = -15
Divide by -5 x = 3
You Try!
1. -2(5x) – 20 = 5x – (-10) 2. x – 4 + (-3x) = 3 – (-2) + x
Part 4: Distributive Property
Use the Distributive Property when there is a number being __________________ by a polynomial.
Make sure you multiply the number by _________________ in the polynomial.
Then you can _________________________________ and solve for x.
Example: 2(x – 5) + 2 = 12
Distribute 2x – 10 + 2 = 12
Combine like terms 2x – 8 = 12
Add 8 2x = 20
Divide by 2 x = 10
You Try!
1. 3(x + 2) – 4 = 9 2. 4(3x – 1) = 5(x + 3) – 5
Part 5: Vocabulary and Math Sentences
Mathematical operations can be expressed by several difference vocabulary words.
For example, ___________________________ means __________________ and _______________ means ________________________
You are going to practice writing mathematical sentences as expressions and vice versa.
Example: A number subtracted by eleven : x – 11
Example: 3x + 4 = 12 : Three times a number plus four is twelve
You Try!
Write each sentence as an expression
1. The sum of five and a number 2. The product of six and a variable
3. Seven subtracted by a number is nine 4. Four times a number minus one is 70
You Try!
Write each expression as a sentence
1. 5x – 2 2. x + 13 = 4
3. x – 8 = 2x + 1 4. 3(x – 1) = 9
Part 6: Word Problems
Being able to solve word problems in math is very important.
It is the closest you will come to ________________________________________
Example: If three is added to twice a number, the result is 17. What is the number?
3 + 2x = 17
2x – 14
x = 7
You Try!
1. Three times a number, increased by one is twenty-five. What is the number?
2. Four times a number, plus three is equal to twice the number, increased by nine. What is the number?
Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 2: INEQUALITIES
Part 1: Graphing Inequalities
In this unit, we learned how to graph inequalities on ______________________________ There are 2 things you must take care of:
What type of circle to use (_____________________________________________) Which way to draw the arrow (__________________________________) Circle:
Open: ____________________ Closed: ____________________
Arrow: Left: ____________________ Right: ____________________
Example: x < 2
g ≥ -4 You Try!
1. m ≤ 5
2. w > 0
3. -7 > d (be careful)
Going Backwards!
Example: s ≤ -2
h > -2
You Try!
1. 2.
____________ _____________
3.
_____________
Part 2: Solving Inequalities
You solve inequalities very similar to solving _____________________
First, _____________________________________, then solve for the variable.
To finish the problem, GRAPH the inequality.
BE CAREFUL! Whenever to multiply or divide by a ________________________________, you must ________________ the inequality
Example: -2x + 5 ≥ 3
Subtract 5 -2x ≥ -2
Divide by -2 (_________________________) x ≤ 1
You Try!
1. 3(x + 1) < 15
2. 4x + 18 ≥ 2
3. -4x – 5 < 15
4. 3x – (-2) > 5x – 12
__________________________________________________________________________________________
Part 3: Word Problems
Inequality word problems use a lot of vocabulary.
Here is a list of the words that go with each symbol:
¿ More Than Above Greater Than
¿ Less Than Below Fewer Than
≥ At Least Minimum No Less Than
≤ At Most Maximum No More Than
Exceeds Higher Than Over
Shorter Lower Than Under
Example: The sum of five times a number and two times and number is at least fourteen.
5x + 2x ≥ 14
7x ≥ 14
x ≥ 2
You Try!
1. The product of a number and three is less than negative twelve
2. Five less than a number is under eleven
3. Negative six times a number is at most forty-eight
__________________________________________________________________________________________
Part 4: Compound Inequalities
Compound Inequalities occur when you have 2 different inequalities written ___________________ and graphed on the __________ number line.
Example: x<2∨x>4
x≥1∧x ≤6
You Try!
Graph the following compound inequalities
1. x<4∨x ≥8
2. x>2∧x<6
Solving Compound Inequalities
________________________ the inequality into 2 different inequalities.
________________ the term in the middle
Example: -1 < x + 9 < 17
Separate -1 < x + 9 and x + 9 < 17
Solve -10 < x and x < 8
You Try!
1. 12<2 x+2≤20
2. 3 x−1≤5<4 x+13
Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 3: STATISTICS
Part 1: Dot Plots
A Dot Plot is a statistical chart containing ___________________________ plotted on a simple scale
To make a dot plot, place a _______ on top of the number every time the data point shows up in the set.
Example:
You Try!
Students from River City High School were randomly selected and asked, “How many pets do you currently own?” The results are recorded below:
__________________________________________________________________________________________
Part 2: Histograms
A Histogram is a statistical chart that shows the ____________________________ of data.
To draw a histogram, _________________ how many data points fall into each __________________ and then draw a bar to represent the frequency.
Example:
You Try!
The following data represents the number of tweets per week.
61 59 66 69 4752 16 13 50 6344 53 49 40 50
__________________________________________________________________________________________
Part 3: 5 Number Summary
___________________ – the _________________ number in the data set
________ – The _______________ of the ______________________ of the data
________ – The _______________ of the data
________ – The _______________ of the _______________________ of the data
_______________ – the _______________ number in the data set
Example: {0, 3, 6, 8, 10, 12, 14}
Minimum – 0 Q1 – 3 Q2 – 8 Q3 – 12 Maximum – 14
You Try! {8, 13, 15, 21, 23, 33, 44}
Minimum –
Q1 –
Q2 –
Q3 –
Maximum –
0123456789
10
Social Media Histogram (TWITTER)
Number of Tweets
Freq
uenc
y
0 – 15 16 – 30 31 – 45 46 – 60 61 – 75
If the data is not in order from lowest to highest, you must put the data in ________________ first
If 2 numbers are in the middle of the data, to find the median, find the _________________ of the 2 numbers.
Example: {19, 2, 13, 8, 0, 5, 3, 2}
Put data in order: {0, 2, 2, 3, 5, 8, 13, 19}
Minimum – 0 Q1 – 2 (Average of 2 and 2) Q2 – 4 (Average of 3 and 5)
Q3 – 11 (Average of 8 and 13) Maximum – 19
You Try! {5, 12, 7, 4, 3, 10, 1, 4}
Minimum –
Q1 –
Q2 –
Q3 –
Maximum –
__________________________________________________________________________________________
Part 4: Outliers
Some data sets include numbers that are either ______________________ or ________________ and don’t fit in with the other numbers in the set.
These are called ___________________
To find any outliers, follow these steps:
1. Find the __________________________________
2. Find the _______________________________________ which is the difference between __________ and _________
3. Multiply the IQR by _________ to find the “magic number”
4. If any numbers in the data are _____________________ Q1 minus the magic number or _____________________ Q3 plus the magic number, they are _______________
Example: {0, 15, 16, 17, 18, 20, 55}
Minimum – 0 Q1 – 15 Q2 – 17 Q3 – 20 Maximum – 55
IQR is 20 – 15, so the IQR is 5.
5 x 1.5 = 7.5 Q1: 15 – 7.5 = 7.5
Q3: 20 + 7.5 = 27.5
0 is less than 7.5 and 55 is more than 27.5, therefore 0 and 55 are outliers
You Try!
{2, 15, 19, 20, 21, 22, 34}
Minimum: Q1: Q2: Q3: Maximum:
IQR: _______
Outliers? _________
__________________________________________________________________________________________
Part 5: Box and Whisker Plots
Box and Whisker plots are a visual representation of the _________________________________
The whiskers are created from the ___________________ and the ____________________
The box is made from __________, __________(the median) and __________.
Example:
You Try! Make a box and whisker plot of the following data:
{60, 60, 62, 63, 63, 66, 66, 66, 67, 68, 69, 69, 69, 69, 70, 70}
Minimum: Q1: Q2: Q3: Maximum:
__________________________________________________________________________________________
Part 6: Stem and Leaf Plots
A plot where each data value is split into a “____________" (usually the last digit) and a “__________" (the other digits).
If any data points are only single digits, use _______ as the stem.
Example:
{22, 26, 27, 31, 33, 35, 42, 44, 46, 57, 58, 59, 61, 63, 64, 65, 67}
You Try!
Make a stem and leaf plot for the following data:
60 61 62 63 64 65 66 67 68 69 70
{12, 45, 23, 44, 18, 33, 9, 51, 17, 23, 43, 33, 30, 13, 10, 7, 40}
__________________________________________________________________________________________
Part 7: Double Stem and Leaf Plots
Double Stem and Leaf Plots – Allow you to plot 2 different sets of data on one stem and leaf plot.
One set of data will be written on the _____________, and the other set will be written on the ________.
The _____________ will be written in the _____________
Example:
Example: The following data represents the ages of 10 Democrats and 10 Republicans in the Senate.
You Try! Make a double stem and leaf plot for the following data:
Males:
9 6 12 11 56 43 38 14 26 13
Females:
18 12 22 45 39 10 8 13 21 27
7 11 10 18 19 32 20 23 16 16
__________________________________________________________________________________________
Males Stem Females
Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 4: FUNCTIONS
Part 1: Linear Input/Output
Input – the numbers ___________________________ to an equation
Output – the __________________ you get after plugging in a number
To finish the table, follow the rule
Example: Rule: Add 4
You Try!
1. Subtract 10 2. Add 4
You can also fill in a table based on an ________________________
Plug in the _______________________ to find the Y value.
Example: y = 3x – 2
You Try!
1. y = x + 8 2. y = 4x + 1
__________________________________________________________________________________________
Part 2: Quadratic Input/Output
Quadratic Function: ____________________
A quadratic function is shaped like a U and is called a ____________________
Example: x2+2x+3
You Try!
1. x2+5 2. x2−3 x+2
__________________________________________________________________________________________
Part 3: Linear Regression
Linear Regression lines follow a _________________________ pattern.
To find the linear regression
1. Plug in table to your calculator (____________________________)
2. Click _____________ and then go over to _____________ and choose _________________
Hit __________. The calculator will give you the linear regression in the format ____________
Example:
Answer: y = 1.05x + 22.32
You Try!
____________________________
__________________________________________________________________________________________
Part 4: Quadratic Regression
Quadratic Regression lines follow a ______________________________ pattern.
To find the quadratic regression
1. Plug in table to your calculator (___________________________)
2. Click ___________ and then go over to __________ and choose _________________
Hit __________. The calculator will give you the linear regression in the format ____________________
Example:
Answer: y = 0.02x 2 – 1.36x + 14.06
You Try!
____________________________
__________________________________________________________________________________________
Part 5: Exponential Regression
Exponential Regression lines will either display exponential _____________ or exponential _________.
To find the exponential regression
1. Plug in table to your calculator (____________________)
2. Click ___________ and then go over to ____________ and choose ________________
Hit ___________. The calculator will give you the linear regression in the format ___________
Example:
Answer: y = 5 ∙ 2 x
You Try!
____________________________
__________________________________________________________________________________________
Part 6: Correlation/R2
Correlation – how ________________ or how ______________ a regression line is
If the correlation/R2 value is close to _______, the regression line is _________________
If the correlation/R2 value is close to _______, the regression line is _____________
Example:
Determine if the regression line for each data set is strong or weak.
1. Linear Regression 2. Quadratic Regression
R2 = 0.9275 Close to 1 STRONG R2 = 0.1111 Close to 0 WEAK
You Try!
1. Exponential Regression
____________________________
2. Linear Regression
___________________________
Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 5: 4 FORMS AND PYTHAGOREAN THEOREM
Part 1: 4 Forms
The 4 Forms of a linear expression are:
1. _________________________________________
2. _________________________________________
3. ______________________
4. ______________________
When converting from the verbal description to the algebraic equation, use the formula _____________
m is the ________________
b is the _____________________________ or __________________________
Example: Sally needs to buy a set of paint brushes and some paint for a picture. The brushes cost $6.98 and the paints are $1.29 each.
Slope: $1.29 Y-Intercept: $6.98 Equation: y = 1.29x + 6.98
You Try!
1. Sheila takes a ballet class. She buys a tutu for $34 and pays $15 per hour for the class.
2. Franklin buys a turtle for $14 at Pet’s Mart. He also has to buy food, which is $2.99 per pound.
__________________________________________________________________________________________
Part 2: Pythagorean Theorem
The Pythagorean Theorem is used to find __________________________________________ in ________________ triangles.
The theorem: _______________________________
a and b are the _______________
c must be the _____________________________
Example
a2 + b2 = c2 62 + 52 = x2
36 + 25 = x2
61 = x2 √61=x 7.8 = x
You try:
X = ________________
__________________________________________________________________________________________
Part 3: Converse to the Pythagorean Theorem
You can also use the Pythagorean Theorem to determine if a triangle is a _________________ triangle
If you perform the Pythagorean Theorem and the equation is ______________, then the triangle ______ a right triangle
If you perform the Pythagorean Theorem and the equation is _______________________, then the triangle is ___________ a right triangle
Example: Side lengths: 4, 12, 10
a2 + b2 = c2 42 + 102 = 122 16 + 100 = 144 116 = 144 Since 144 is not equal to 116, the triangle is NOT RIGHT
You try: Is the triangle a right triangle? Yes or No?
1. Side lengths: 9, 40, 41 2. Side lengths: 8, 7, 5
_____________ _____________
__________________________________________________________________________________________
Part 4: Word Problems
When solving word problems with the Pythagorean Theorem, it is very important that you put the right _____________________ on the right _____________ of the triangle.
Example: Jerry climbed up to the top of a slide and then slid down. He had so much fun that he wants to do it again. If he climbed up 6ft and then slid down the 13ft long slide, how far does he have to walk to get back to the ladder where he can climb back up?
You try!
1. The bottom of a 13-foot straight ladder is set into the ground 5 feet away from a wall. When the top of the ladder is leaned against the wall, what is the distance above the ground it will reach?
2. David leaves the house to go to school. He walks 200 m west and 125 m north. Calculate how far he is from the starting point.
6ft13ft
62
+ b2
= 132
36 + b2
= 169
b2
= 133
Intro to HS Math Final Exam Review Name: ______________________________________
UNIT 6: VOLUME AND POLYNOMIALS
Part 1: Volume of Prisms and Cubes
Volume of ________________________________________ and _____________
Formula: _______________________
Example: V=5 cm∙4cm ∙10cm
V=200cm3
Don’t forget your UNITS!
You Try!
Find the Volume of the Rectangular Prism and Cube
1. 2.
Volume of _____________________________________
Formula: ____________________
Example: V=12∙4 cm∙6cm∙8cm
V=96cm3
Don’t forget your UNITS!
You Try!
Find the Volume of the Triangular Prism
Part 2: Volume of Cylinders and Spheres
Volume of __________________________
Formula: _____________________
If you were given the diameter of the circle, divide it in ___________ to find the ________________
Example: V=π ∙(5cm)2 ∙12cm
V=942.5cm3
Don’t forget your UNITS!
You Try!
Find the volume of the Cylinders
1. 2.
Volume of __________________
Formula: ________________
If you were given the diameter of the sphere, divide it in ___________ to find the ________________
Example: V= 43∙ π ∙¿¿
V=268.1¿3
Don’t forget your UNITS!
You Try!
Find the volume of the Sphere
Part 3: Volume of Pyramids and Cones
Volume of ________________________
Formula: __________________
Example: V=13∙8∈∙7∈∙6∈¿
V=112¿3
Don’t forget your UNITS!
You Try!
Find the volume of the Pyramid
Volume of _______________
Formula: ___________________
Example: V=13∙ π ∙(8cm)2 ∙18cm
V=1206.4 cm3
Don’t forget your UNITS!
You Try!
Find the volume of the Cone
__________________________________________________________________________________________
Part 4: Naming Polynomials by Degree
The degree of a polynomial is the _________________________________ in the polynomial
Example: y=4 x3−5 x+1 y=5−3 x+5 x2−4 x7
Degree = 3 Degree = 7
You Try!
1. −2 x5+6 x3−2 x+9 = _____________________________
2. 9 x3+3x6+x−8 = _____________________________
3. 5 x5+x7−4 x9+8 x10+12 = _____________________________
4. x3+ x5 = _____________________________
__________________________________________________________________________________________
Part 5: Naming Polynomials by the Number of Terms
Polynomials are made up of __________________________________
They can be named based on how many terms they have
1 Term: ___________________
2 Terms: ___________________
3 Terms: ___________________
4 or more Terms: ___________________
Example:
1. 2 x2+4 x 2 TERMS – _________________
2. 4 x3−3 x2+5 x−1 4 TERMS – _________________
3. −4 x 1 TERM – _________________
4. 2 x5−8 x2−6 3 TERMS – _________________
You Try!
1. −3 x2 = _________________
2. 5 x3−8 x2+2 x= _________________
3. −9 x5+2x4−5 x3+6 x2−7 x+1 = _________________
4. 2 x4+9 x2 = _________________
Name the following polynomials by degree and number of terms
Example: y=4 x3−7 x2+5x
TRINOMIAL, DEGREE: 3
You Try!
1. y=7+2x5 2. y=6x3+4 x2−10x+3
_________________________________ _________________________________
Part 6: Adding and Subtracting Polynomials
When you are adding and subtracting polynomials, you must:
_____________________________________- only!
Make sure to ________________________ the ___________________ when subtracting
Example:
(4x2 + 9x – 6) + (7x2 – 2x – 1) = 11x2 + 7x – 7
(3x2 + 5x – 8) – (5x2 – 4x + 6) = -2x2 + 9x – 14
You Try!
1. (4x2 + 9x – 6) + (7x2 – 2x – 1) = ________________________________________
2. (3x2 + 5x – 8) – (5x2 – 4x + 6) = ________________________________________
3. (3x –+2) – (4x + 1) + (7x – 9) = ________________________________________