Student: ______________________

Richard Montgomery HS

2016-17

ALGEBRA 2

Linear Functions Review

Page 2 of 24

WHAT YOU WILL LEARN

In this packet, you will learn how to:

1. Calculate slope from ordered pairs using the slope formula.

2. Write linear equations from tables, graphs, or pairs of points.

3. Graph, solve and interpret linear functions expressed in slope-

intercept form and point-slope form.

4. Solve systems of linear equations using graphing, elimination,

linear combination and substitution.

5. Graph and interpret piecewise linear functions.

6. Determine the equation for a piecewise function from its graph.

Schedule - *Subject to Change*

Day # Objective Lesson pages Homework

1 Slope-Intercept Form Review Page 3 – 6 Page 7

2 Point-slope form Page 8 – 11 Page 12

3 Linear Systems of Equations Page 13 – 16 Page 17

4 Linear functions application

activity Page 18 Page 19

5 Piecewise Linear Functions Pages 20 – 23 Page 24

Page 3 of 24

DAY 1

SLOPE-INTERCEPT REVIEW

We can still think of slope as 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑦

𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑥 or

∆ 𝑦

∆ 𝑥.

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:

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.

Page 4 of 24

Two Special Cases

Slope = ______________________ Slope = ______________________

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 = ∆ 𝑦

∆ 𝑥=

𝑦2−𝑦1

𝑥2−𝑥1

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 ∆ 𝑦

∆ 𝑥=

=

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.

Page 5 of 24

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 _____________ and the number

in place of the letter b represents the __________________.

How to Graph a Line in Slope-Intercept Form

Example: Graph the function 𝑦 =1

2𝑥 – 3.

What is the y-intercept? __________

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? __________

What is the slope?

_____________________________

What is the equation for this function?

________________________________

Page 6 of 24

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 𝑚 =𝑦2−𝑦1𝑥2−𝑥1

.

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

Step 3: Write the slope-intercept form.

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.

Page 7 of 24

Homework Day 1

Slope-Intercept Form

Graph these linear equations on the graph provided.

1. 𝑦 = −2

3𝑥 + 4

y-intercept: ___________ Slope: ____________

2. 𝑦 = 𝑥

y-intercept: ___________ Slope: ____________

Determine the slope, y-intercept and equations for each of the following linear functions.

3. 4. 5.

y-intercept: _____________

Slope: ____________

Equation: ________________

y-intercept: _____________

Slope: ____________

Equation: ________________

y-intercept: _____________

Slope: ____________

Equation: ________________

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)

Page 8 of 24

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 = −1

3 that contains

the point (5, 2).

We can use the point-slope form to graph a linear equation.

3. Graph 𝑦 − 1 = −5

3(𝑥 − 2)

4. Graph 𝑦 + 3 = (𝑥 + 2)

5. Graph 𝑦 − 4 = 0(𝑥 + 6)

Page 9 of 24

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

Example #6:

7. Find the equation for the

following graph:

8. Find the equation for the

following graph:

9. Find the equation for the

following graph:

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

Page 10 of 24

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 =1

2(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.

Example 3: Graph the line 3𝑥 + 2𝑦 = 8.

This line is in standard form. We need to solve for y first

before graphing

Page 11 of 24

1) Mana and her friends are going to see the first show of the movie,

Guardians of the Galaxy, Vol. 2. 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?

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

What is the starting value of the table?

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

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

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

f(2) = ___________

https://i.annihil.us/u/prod/marvel/i/mg/9/c0/58d54d2cdc863/portrait_incredible.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.

Page 12 of 24

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)

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

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

5. 𝑦 = −3

4𝑥+ 1

6. 𝑦 + 4 =2

3(𝑥 + 1)

1. 2.

Slope: _______ y-intercept: __________

Equation: ___________________

Meaning of Slope:

Meaning of y-intercept:

Slope: _______ y-intercept: __________

Equation: ___________________

Meaning of Slope:

Meaning of y-intercept:

Page 13 of 24

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?

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 + 4)𝑦 = 4𝑥 + 8

Solutions: _______ Solutions: _______ Solutions: _______

Number of solutions: ______ Number of solutions: ______ Number of solutions: ______

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.

Page 14 of 24

Example 1: Solve the following system of

equations:

{3𝑥 − 2𝑦 = 122𝑥 + 2𝑦 = −2

Steps: 1. Determine which variable to

eliminate.

2. Add or subtract the equations, then solve

for the remaining variable.

3. Replace the answer you just found into one

of your original equations to find the value of

the other variable.

Example 2: Solve the following system of

equations:

{4𝑥 + 5𝑦 = 74𝑥 − 𝑦 = 1

Steps: 1. Determine which variable to

eliminate.

2. Add or subtract the equations, then solve

for the remaining variable.

3. Replace the answer you just found into one

of your original equations to find the value of

the other variable.

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.

Page 15 of 24

Example 1: Solve by the Linear Combination

Method.

{6𝑥 + 3𝑦 = −152𝑥 + 6𝑦 = 10

Steps: 1. Determine which variable to

eliminate.

2. Multiply one of the equations by a constant

so that we can eliminate one variable.

3. Add or subtract both equations and solve

for the remaining variable.

4. Replace the answer you just found into one

of your original equations to find the value of

the other variable.

Example 2: Solve by the Linear Combination

Method.

{5𝑥 + 3𝑦 = −103𝑥 + 5𝑦 = −6

Steps: 1. Determine which variable to

eliminate.

2. Multiply both equations by a constant so

that we can eliminate one variable.

3. Add or subtract both equations and solve

for the remaining variable.

4. Replace the answer you just found into one

of your original equations to find the value of

the other variable.

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.

Page 16 of 24

Example 1: Solve the following system of

equations:

{𝑥 + 𝑦 = 5𝑦 = 3 + 𝑥

Steps: 1. Solve for one variable in one

equation.

2. Substitute for that variable in the other

equation.

3. Solve for the remaining variable.

4. Replace the answer you just found into one

of your equations to find the value of the

other variable.

Example 2: Solve the following system of

equations:

{𝑥 + 3𝑦 = 74𝑥 − 2𝑦 = 0

Steps: 1. Solve for one variable in one

equation.

2. Substitute for that variable in the other

equation.

3. Solve for the remaining variable.

4. Replace the answer you just found into one

of your equations to find the value of the other

variable.

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.

Page 17 of 24

Homework Day 3

Solving systems of linear equations

Solve the following systems:

1. {3𝑥 + 2𝑦 = −14𝑥 + 2𝑦 = −6

2. {2𝑥 − 3𝑦 = −1𝑦 = 𝑥 − 1

3. {𝑦 = 6𝑥 − 11

−2𝑥 − 3𝑦 = −7 4. {

2𝑚 + 3𝑛 = 4−𝑚 + 2𝑛 = 5

Page 18 of 24

DAY 4

APPLYING LINEAR EQUATIONS

Today you will work in groups to solve the following problem. Record your answers on your

group’s answer sheet.

Page 19 of 24

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.

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.

Determine how many hours the cost would be the same for both parking garages. What is

the cost for this amount of time?

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?

Page 20 of 24

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 ___________________ 1.39, 𝑖𝑓 ___________________ 1.56, 𝑖𝑓 ___________________ 1.73, 𝑖𝑓 ___________________ 1.90, 𝑖𝑓 ___________________

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

Price for a letter that weighs 2.4 ounces:

Price of a letter that weighs 0.79 ounces:

Price of a letter that weighs 4 ounces:

Page 21 of 24

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.

Let x = your dog’s weight (in pounds).

Determine the function for this situation:

𝑓(𝑥) = {

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?

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(𝑥) = {

How much would it cost if you order

35 shirts?

How much would it cost if you order

70 shirts?

Page 22 of 24

Graph the following on the grids.

1. 𝑓(𝑥) = {2𝑥, 𝑖𝑓 𝑥 < 2

−𝑥 + 7, 𝑖𝑓 𝑥 ≥ 2 2. 𝑓(𝑥) = {

−1

2𝑥 −

5

2, 𝑖𝑓 𝑥 ≤ −1

3𝑥 − 4, 𝑖𝑓 𝑥 > −1

Boundary Points: Boundary Points:

3. 𝑓(𝑥) = {𝑥 + 1, 𝑖𝑓 𝑥 ≤ 43

2𝑥 − 6, 𝑖𝑓 𝑥 > 4

4. 𝑓(𝑥) = {−3, 𝑖𝑓 𝑥 < 0

−𝑥 + 2, 𝑖𝑓 𝑥 ≥ 0

Boundary Points: Boundary Points:

When graphing piecewise functions, we need to pay attention to the

restrictions on the domain of each function. We can start by figuring out the

“boundary points” of each piece, and then using what we know about the slope

and points on the line to complete the graph.

Page 23 of 24

Example #1: Write equations for the piecewise function

whose graph is shown on the right.

Example #2: Find the equation for the piecewise

function graphed on the right.

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 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Page 24 of 24

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) = {

1

2𝑥 − 4, 𝑖𝑓 𝑥 ≤ −2

3 − 2𝑥, 𝑖𝑓 𝑥 > −2

Evaluate each function for the given value of x:

1. f(2)

2. f(–4) 3. g(–1) 4. g(3)

5. g(7) 6. h(–2) 7. h(–1) 8. h(6)

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.

11. 10.