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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.