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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?


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

__________________________________________________________________________________________


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


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


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

___________________________


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

_____________ _____________

__________________________________________________________________________________________


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


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


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


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


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


You Try!

Find the volume of the Pyramid

Volume of _______________

Formula: ___________________

Example: V=13∙ π ∙(8cm)2 ∙18cm

V=1206.4 cm3


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) = ________________________________________

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