Linears Review Packet KEY - Math with Mrs. Pierce

Student: ______________________

Richard Montgomery HS 2018-2019

Honors Algebra 2

Linear Functions Review KEY

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DAY 1 SLOPE-INTERCEPT REVIEW

We can still think of slope as

or

∆

∆ .

Another way that slope is sometimes described is with the phrase “Rise Over Run,” as in the example below.

For graphs that increase, the slope is positive. For graphs that decrease, the slope is negative. Examples: Determine the slopes of the following lines:

12

41

4 22

1 5

3

In previous lessons, we have used the rate of change of linear functions to write equations. In this lesson, we will use rates of change to help us graph linear functions. With linear equations, the rate of change is also called the slope of the linear equation.

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Two Special Cases

Slope = 0 Slope = Undefined Sometimes you will be asked to find the slope (or rate of change) from coordinate points, or from given values in various forms. When finding slope given two points, the formula is written like this:

Slope = ∆

∆

Example: Find the slope of the line between the coordinate points (3, 6) and (7, 8). In this problem, we can say that (x1, y1) = (3, 6) and (x2, y2) = (7, 8).

Therefore, our slope equals ∆

∆

Important note: Remember to simplify your answers.

With linear equations, it is possible to extract useful information from the equation that will help us graph it quickly with minimal calculations. However, this is only possible when the equation is in a particular form.

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Definition: A linear equation is said to be in slope-intercept form when it is written in the following format:

𝑦 𝑚𝑥 𝑏

In this format, the number in place of the letter m represents the slope and the number in place

of the letter b represents the y-intercept.

How to Graph a Line in Slope-Intercept Form

Example: Graph the function 𝑦 𝑥 – 3.

What is the y-intercept? –3

What is the slope?

Graph the y-intercept on the grid, then use the

slope to fill in other points on the line.

Connect the line to finish the problem.

How to determine an equation from a line Example: Look at the graph on the right.

What is the y-intercept? –1

What is the slope?

31

3

What is the equation for this function?

𝑦 3𝑥 1

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Example: Write the equation of the line that passes through (–2, 3), and (–6, 1).

Step 1: Find the slope of the line containing the points. Using 𝑚𝑦2 𝑦1𝑥2 𝑥1

.

𝑚1 3

6 224

12

Step 2: You know the slope now and two points. Choose one point and find the y-intercept.

312

2 𝑏

3 1 𝑏

4 𝑏

Step 3: Write the slope-intercept form.

𝑦12

𝑥 4

We have practiced how to find the equation of a linear function using a variety of different kinds of information. To conclude, we look at how to determine the equation of a line based on only two points.

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Homework Day 1 Slope-Intercept Form

Graph these linear equations on the graph provided.

1. 𝑦 𝑥 4

y-intercept: 4 Slope:

2. 𝑦 𝑥

y-intercept: 0 Slope: 1

Determine the slope, y-intercept and equations for each of the following linear functions. 3. 4. 5. y-intercept: –4

Slope:

Equation: 𝑦 𝑥 4

y-intercept: 2

Slope: 4

Equation: 𝑦 4𝑥 2

y-intercept: –3

Slope:

Equation: 𝑦 𝑥 3

Write the equation of the line that passes through each pair of points.

6. (1, 1) and (0, –3) 7. (–1, 6) and (3, 4)

𝑚3 1

0 141

4

𝑦 4𝑥 3

𝑚4 6

3 12

412

6 1/2 1 𝑏 6 0.5 𝑏

5.5 𝑏

𝑦12

𝑥 5.5

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DAY 2 POINT-SLOPE FORM

Previously, we have dealt with the slope-intercept form of a line, which uses the formula

𝑦 𝑚𝑥 𝑏. Example #1: Write an equation in slope-intercept form for the line with slope 𝑚 4 with the y-intercept (0, 2). Now we will look at another way of writing the equation of a line.

Example #2: Write an equation in point-slope form for the line with slope 𝑚 that contains the point (5, 2).

𝑦 213

𝑥 5

We can use the point-slope form to graph a linear equation. 3. Graph 𝑦 1 𝑥 2

4. Graph 𝑦 3 𝑥 2

5. Graph 𝑦 4 0 𝑥 6

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We can also find the equation of a line in point-slope form using two points. Example #6:

𝑦 2 3 𝑥 4 7. Find the equation for the

following graph:

8. Find the equation for the following graph:

9. Find the equation for the following graph:

𝑦 3 1 𝑥 4 𝑦 3 3 𝑥 2 𝑦 2 0 NOTE: These equations are not unique. That is, there is more than one possible equation in point-slope form for all of these graphs. So far, we have seen three different formulas for the equation of a line: Sometimes, we need to convert from one form to another. Usually, we do this so that we can graph the line.

Slope-Intercept Form: y = mx + b Point-Slope Form: y – y1 = m(x – x1)

Standard Form: Ax + By = C

4 26 4

62

3

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Example 1: Graph the line 𝑦 2𝑥 5. Since we already have this in slope-intercept form, we can graph this right away.

Example 2: Graph the line 𝑦 9 𝑥 20 .

This line is in point-slope form. We can normally graph these directly, but since the point (20, 9) is not on our graph, we should solve for y first.

𝑦 912

𝑥 10

𝑦12

𝑥 1

Example 3: Graph the line 3𝑥 2𝑦 8. This line is in standard form. We need to solve for y first before graphing

2𝑦 3𝑥 8

𝑦3𝑥 8

2

𝑦32

𝑥 4

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1) Mana and her friends are going to see the first show of the movie, Avengers: Endgame. They spent money on movie tickets and snacks. The table below shows the number of tickets and the total amount of money paid.

Number of tickets (x) Total Amount Paid f(x) 0 15

3 51

6 87

9 123

What is the rate of change of the table?

𝑚363

12

What does the rate of change mean in the context of the problem?

Each ticket costs $12.

What is the starting value of the table?

15

What does the starting value mean in the context of the problem?

The snacks cost $15.

Write an explicit equation to represent the table. (𝑦 𝑚𝑥 𝑏)

𝑦 12𝑥 15

Fill in the blank, then identify what the mathematical statement means.

f(2) = 12 2 15 24 15 39 The total cost for snacks and two tickets is $39.

https://pbs.twimg.com/media/D1pKLzbUYAAb0la.jpg

To conclude this review lesson, we will remind ourselves of how to interpret the slope and y-intercept of a linear function in the context of a word problem.

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Homework Day 2 Point-Slope Form

For #1-4, write the equation of the line with the given slope that goes through the given point. (Hint: use point-slope form) 1. Slope = 5, through (8, 6) 2. Slope = –11, through (3.4, –10)

𝑦 6 5 𝑥 8 𝑦 10 11 𝑥 3.4

3. Slope = 74, through (17, 0) 4. Slope = –1, through (–6, 27)

𝑦74

𝑥 17 𝑦 27 1 𝑥 6

For #5-6, graph both equations on the grid on the right.

5. 𝑦 34

𝑥 1

6. 𝑦 4 23

𝑥 1

1. 2.

Slope: 5 y-intercept: 90

Equation: 𝑦 5𝑥 90 Meaning of Slope: The savings account loses $5 per week. Meaning of y-intercept: There is $90 in the account at the start.

Slope: y-intercept: 0

Equation: 𝑦 𝑥

Meaning of Slope: It costs $3 to play 2 games at the arcade (or $1.50 per game). Meaning of y-intercept: It does not cost anything to go to the arcade.

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DAY 3 SOLVING SYSTEMS OF LINEAR EQUATIONS

A solution of a system of linear equations is an ordered pair (x, y) that satisfies BOTH equations. For each problem, determine whether the given ordered pair is a solution to the system.

Is (3, –2) a solution to this system? Is (0, 7) a solution to this system?

Yes, because 3 3 5 3 2

and 3 3 5 3 2

No, because 4 0 3 0 3 3 7

Graph the following systems of equations on the grids provided.

Example 1: 𝑦 3𝑥 9𝑦 2𝑥 6 Example 2:

𝑦 5𝑥 4𝑦 5𝑥 2 Example 3:

𝑦 2 2𝑥 4𝑦 4𝑥 8

Solutions: (3, 0) Solutions: None Solutions: All real numbers Number of solutions: 1 Number of solutions: 0 Number of solutions: Infinite

A collection of two or more linear equations is called a linear system of equations, or simply a linear system. This lesson covers features of linear systems and methods used to solve them.

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Example 1: Solve the following system of equations:

3𝑥 2𝑦 122𝑥 2𝑦 2

Steps: 1. Determine which variable to eliminate. y, since both y terms have a coefficient

of 2. 2. Add or subtract the equations, then solve for the remaining variable.

3𝑥 2𝑦 12 2𝑥 2𝑦 2

5𝑥 10 𝑥 2 3. Replace the answer you just found into one of your original equations to find the value of the other variable.

2 2 2𝑦 2

4 2𝑦 2

2𝑦 6

𝑦 3 So, our solution is (2, –3).


4𝑥 5𝑦 74𝑥 𝑦 1

Steps: 1. Determine which variable to eliminate. x, since both x terms have a coefficient

of 4. 2. Add or subtract the equations, then solve for the remaining variable.

4𝑥 5𝑦 7 4𝑥 𝑦 1

6𝑦 6 𝑦 1 3. Replace the answer you just found into one of your original equations to find the value of the other variable.

4𝑥 5 1 7

4𝑥 5 7

4𝑥 2

𝑥 0.5 So, our solution is (0.5, 1).

Previously, we have solved systems of equations by graphing the equations. On this page, we will start to look at ways to solve systems of equations algebraically. The purpose of elimination method is to “eliminate” or “cancel out” one of the variables. Elimination problems will be written in standard form: ax + by = c.

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Example 1: Solve by the Linear Combination Method.

6𝑥 3𝑦 152𝑥 6𝑦 10

2. Multiply one of the equations by a constant so that we can eliminate one variable.

3 2𝑥 6𝑦 3 10 6𝑥 18𝑦 30

3. Add or subtract both equations and solve for the remaining variable.

6𝑥 3𝑦 15 6𝑥 18𝑦 30

15𝑦 45 𝑦 3 4. Replace the answer you just found into one of your original equations to find the value of the other variable.

2𝑥 6 3 10

2𝑥 18 10

2𝑥 8

𝑥 4 So, our solution is (–4, 3).

Example 2: Solve by the Linear Combination Method.

5𝑥 3𝑦 103𝑥 5𝑦 6

2. Multiply both equations by a constant so that we can eliminate one variable.

5 5𝑥 3𝑦 5 10 25𝑥 15𝑦 50

3 3𝑥 5𝑦 3 6

9𝑥 15𝑦 18 3. Add or subtract both equations and solve for the remaining variable.

25𝑥 15𝑦 50 9𝑥 15𝑦 18

16𝑥 32 𝑥 2 4. Replace the answer you just found into one of your original equations to find the value of the other variable.

5 2 3𝑦 10

10 3𝑦 10

3𝑦 0

𝑦 0 So, our solution is (–2, 0).

Sometimes when you cannot use elimination as the first step to solving a system of linear equations you need to preform another step first. This algebraic method is called linear combination. Below are some examples of linear combination.

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𝑥 𝑦 5𝑦 3 𝑥

Steps: 1. Solve for one variable in one equation. In this problem, the second equation is already solved for y. 2. Substitute for that variable in the other equation.

𝑥 3 𝑥 5 3. Solve for the remaining variable.

2𝑥 3 5

2𝑥 2

𝑥 1 4. Replace the answer you just found into one

of your equations to find the value of the other variable.

𝑦 3 1

𝑦 4

So, our solution is (1, 4).


𝑥 3𝑦 7

4𝑥 2𝑦 0

Steps: 1. Solve for one variable in one equation. In the first equation, x has a coefficient of 1. This will make it simpler to solve for x.

𝑥 7 3𝑦 2. Substitute for that variable in the other equation.

4 7 3𝑦 2𝑦 0 3. Solve for the remaining variable.

28 12𝑦 2𝑦 0

28 14𝑦 0

14𝑦 28

𝑦 2 4. Replace the answer you just found into one of your equations to find the value of the other variable.

𝑥 7 3 2

𝑥 7 6

𝑥 1 So, our solution is (1, 2).

Our final method for solving systems of equations involves solving an equation for one of the variables, and then substituting the result into the other equation.

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Homework Day 3 Solving systems of linear equations

1) 2)

Solve the following systems:

1. 3𝑥 2𝑦 14𝑥 2𝑦 6

1𝑥 5

𝑥 5

3 5 2𝑦 1 15 2𝑦 1

2𝑦 14 𝑦 7

2. 2𝑥 3𝑦 1

𝑦 𝑥 1

2𝑥 3 𝑥 1 1 2𝑥 3𝑥 3 1

1𝑥 3 1 1𝑥 4 𝑥 4

𝑦 4 1

𝑦 3

3. 𝑦 6𝑥 112𝑥 3𝑦 7

2𝑥 3 6𝑥 11 7 2𝑥 18𝑥 33 7

20𝑥 33 7 20𝑥 40

𝑥 2

𝑦 6 2 11 𝑦 1

4. 2𝑚 3𝑛 4𝑚 2𝑛 5

7𝑛 14

𝑛 2

2𝑚 3 2 4 2𝑚 6 4

2𝑚 2 𝑚 1

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Homework Day 4 Applications of Systems

1. You are going to a baseball game at Camden Yards. A parking garage down the street charges a flat fee of $10 and $0.50 per hour for parking. A parking garage across the street from the ballpark charges a flat fee of $5 and $1.50 per hour for parking.

For each parking garage, write an equation that represents the total costs for parking.

𝑦 10 0.5𝑥 𝑦 5 1.5𝑥

Solve the system of equations you wrote in part (a). The grid

on the right is provided in case you would like to use graphs.

10 0.5𝑥 5 1.5𝑥 10 5 𝑥

5 𝑥

10 0.5 5 10 2.5 12.5

Determine how many hours the cost would be the same for both parking garages. What is the cost for this amount of time?

It would cost the same ($12.50) for both garages after 5 hours.

2. The Park & Ride Parking Lot charges $7 per day if you have a frequent parker sticker and

$10 per day if you do not. Last Friday there were 35 cars in the lot and they made a total of $290. How many cars had the frequent parker sticker and how many did not?

7𝑥 10𝑦 290 𝑥 𝑦 35

𝑦 35 𝑥

7𝑥 10 35 𝑥 290 7𝑥 350 10𝑥 290

3𝑥 350 290 3𝑥 60 𝑥 20

𝑦 35 20 𝑦 15

20 cars had a frequent parker sticker and 15 did not.

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Day 5 Piecewise Linear Functions

In our next unit, we will be studying piecewise functions, which are functions that are defined in different ways depending on the value of x.

First, let’s begin with word problems. Example #1: Step functions are used widely in everyday life, especially in the world of business. Utilities (gas, electricity, water, etc,) are often billed according to a step function. Here are first-class mail rates for packages from the US Postal Service‛s web site: Let x = the weight of a first-class package First, we need to define a function for this situation:

𝑓 𝑥

⎩⎪⎨

⎪⎧

1.22, 𝑖𝑓 0 𝑥 1 1.39, 𝑖𝑓 1 𝑥 2 1.56, 𝑖𝑓 2 𝑥 3 1.73, 𝑖𝑓 3 𝑥 4 1.90, 𝑖𝑓 4 𝑥 5

Let’s say we want to use the function to determine the following:

Price for a letter that weighs 2.4 ounces:

$1.56

Price of a letter that weighs 0.79 ounces:

$1.22

Price of a letter that weighs 4 ounces:

$1.90

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Example #2: Your favorite dog groomer charges according to your dog’s weight. If your dog is 15 pounds and under, the groomer charges $35. If your dog is between 15 and 40 pounds, she charges $40. If your dog is over 40 pounds, she charges $40, plus an additional $2 for each pound over 40. Let x = your dog’s weight (in pounds).

Determine the function for this situation:

𝑓 𝑥 35, 𝑖𝑓 𝑥 1540, 𝑖𝑓 15 𝑥 4040 2 𝑥 40 , 𝑖𝑓 𝑥 40

How much would it cost for grooming if

your dog weighs 23 pounds? How much would it cost for grooming if

your dog weighs 60 pounds? $40. 40 + 2(60 – 20) = 40 + 2(20) = 40 + 40 = $80 Example #3: You are ordering shirts for an end-of-year celebration. The t-shirt company has the following requirements:

An initial charge of $20 to create the silk screen $17.00 per shirt for orders of 50 or fewer shirts $15.80 per shirt for orders of more than 50 shirts

Let x = the number of t-shirts ordered.

Determine the function for this situation:

𝑓 𝑥20 17𝑥, 𝑖𝑓 𝑥 50

20 15.8𝑥, 𝑖𝑓𝑥 50

How much would it cost if you order

35 shirts? How much would it cost if you order

70 shirts? 20 + 17(35) = 20 + 595 = $615 20 + 15.8(70) = 20 + 1106 = $1126

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Graph the following on the grids.

1. 𝑓 𝑥 2𝑥, 𝑖𝑓 𝑥 2

𝑥 7, 𝑖𝑓 𝑥 2 2. 𝑓 𝑥 𝑥 , 𝑖𝑓 𝑥 1

3𝑥 4, 𝑖𝑓 𝑥 1

Boundary Points: 2 2 4 → 2, 4 → 𝑂𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒

2 7 5 → 2, 5 → 𝐶𝑙𝑜𝑠𝑒𝑑 𝑐𝑖𝑟𝑐𝑙𝑒

Boundary Points: 12

152

12

52

42

2

→ 1, 2 → 𝐶𝑙𝑜𝑠𝑒𝑑 𝑐𝑖𝑟𝑐𝑙𝑒 3 1 4 3 4 7

→ 1, 7 → 𝑂𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒

3. 𝑓 𝑥 𝑥 1, 𝑖𝑓 𝑥 4

𝑥 6, 𝑖𝑓 𝑥 4 4. 𝑓 𝑥

3, 𝑖𝑓 𝑥 0𝑥 2, 𝑖𝑓 𝑥 0

Boundary Points: 4 1 5 → 4, 5 → 𝐶𝑙𝑜𝑠𝑒𝑑 𝑐𝑖𝑟𝑐𝑙𝑒

32

4 6 6 6 0 → 4, 0 → 𝑂𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒

Boundary Points: → 0, 3 → 𝑂𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒

0 2 2 → 0, 2 → 𝐶𝑙𝑜𝑠𝑒𝑑 𝑐𝑖𝑟𝑐𝑙𝑒

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Example #1: Write equations for the piecewise function whose graph is shown on the right.

Slope for the first “piece”: 𝑚 1

y-intercept of the first “piece”: 𝑏 2 Slope for the second “piece”: 𝑚 1

y-intercept of the second “piece”: 𝑏 0

𝑓 𝑥𝑥 2, 𝑖𝑓 𝑥 0

𝑥, 𝑖𝑓 𝑥 0

Example #2: Find the equation for the piecewise function graphed on the right.

Slope for the first “piece”: 𝑚

y-intercept of the first “piece”: 𝑏 4 Slope for the second “piece”: 𝑚 5

Since we don’t see the y-intercept of the second “piece”, we can use point-slope form:

𝑦 1 5 𝑥 4 𝑦 5 𝑥 4 1

𝑓 𝑥14

𝑥 4, 𝑖𝑓 𝑥 4

5 𝑥 4 1, 𝑖𝑓 𝑥 4

When determining the equation for a piecewise function from its graph, keep in mind the different linear forms we have discussed: Slope-intercept form: 𝑦 𝑚𝑥 𝑏 Point-slope form: 𝑦 𝑦 𝑚 𝑥 𝑥

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Homework Day 5 Intro to Piecewise Functions

Use the following function definitions:

𝑓 𝑥 3, 𝑖𝑓 𝑥 02, 𝑖𝑓 𝑥 0 𝑔 𝑥

𝑥 5, 𝑖𝑓 𝑥 32𝑥 1, 𝑖𝑓 𝑥 3 ℎ 𝑥

12

𝑥 4, 𝑖𝑓 𝑥 2

3 2𝑥, 𝑖𝑓 𝑥 2

Evaluate each function for the given value of x: 1. f(2) = 2

2. f(–4) = 3 3. g(–1) = –1 + 5 = 4 4. g(3) = 3 + 5 = 8

5. g(7) = 2(7) – 1 = 13

6. h(–2) = 2 4 = –1 – 4 = –5

7. h(–1) = 3 2 1 = 3 + 2 = 5

8. h(6) = 3 2 6 = 3 – 12 = –9

9) The admission rates at an amusement park are as follows.

Children under 5 years old: free Children at least 5 years and less than 12 years: $10.00 Children at least 12 years and less than 18 years: $25.00 Adults: $35.00

Write a piecewise function that gives the admission price for a given age.

Let x be the age of the customer.

𝑓 𝑥

0, 𝑖𝑓 𝑥 510, 𝑖𝑓 5 𝑥 1225, 𝑖𝑓 12 𝑥 1835, 𝑖𝑓 𝑥 18

11.10.

Linears Review Packet KEY - Math with Mrs. Pierce

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Transcript of Linears Review Packet KEY - Math with Mrs. Pierce