Name: Chapter 2: Linear Relations and Functions

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Lesson 2-1: Relations and Functions Date:

Example 1: Domain and Range

State the domain and range of the relation. Then determine whether the relation is a function. If it is a

function, determine if it is one-to-one, onto, both, or neither.

Name: Chapter 2: Linear Relations and Functions

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A relation in which the domain is a set of individual points is said to be a relation.

When the domain of a relation has an infinite number of elements and can be graphed with a line or

smooth curve, the relation is a relation.

Real-World Example 2

TRANSPORTATION The table shows the average fuel efficiency in miles per gallon for SUVs for several

years. Graph this information and determine whether it represents a function. Is this relation discrete or

continuous?

Example 3: Graph a Relation

Graph 𝑦 = 3𝑥 – 1 and determine the domain and range. Then determine whether the equation is a

function, is one-to-one, onto, both, or neither. State whether it is discrete or continuous.

Example 4: Evaluate a Function

Given 𝑓(𝑥) = 𝑥3 – 3, find

A. B.

x

y

x

y

Name: Chapter 2: Linear Relations and Functions

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Lesson 2-2: Linear Relations and Functions Date:

Relations that have straight line graphs are called relations.

Relations that are not linear are called relations.

A function is a function that can be written in the form 𝑓(𝑥) = 𝑚𝑥 + 𝑏,

where 𝑚 and 𝑏 are real numbers.

Example 1: Identify Linear Functions

A. State whether 𝑔(𝑥) = 2𝑥 – 5 is a linear function. Write yes or no. Explain.

B. State whether 𝑝(𝑥) = 𝑥3 + 2 is a linear function. Write yes or no. Explain.

Real-World Example 2: Evaluate a Linear Function

METEOROLOGY The linear function 𝑓(𝐶) = 1.8𝐶 + 32 can be used to find the number of degrees

Fahrenheit 𝑓(𝐶) that are equivalent to a given number of degrees Celsius 𝐶.

A. On the Celsius scale, normal body temperature is 37°𝐶. What is it in degrees Fahrenheit?

B. There are 100 Celsius degrees between the freezing and boiling points of water and 180 Fahrenheit

degrees between these two points. How many Fahrenheit degrees equal 1 Celsius degree?

Name: Chapter 2: Linear Relations and Functions

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Example 3: Standard Form

Write 𝑦 = 3𝑥 – 9 in standard form. Identify 𝐴, 𝐵, and 𝐶.

The y-coordinate of the point at which a graph crosses the y-axis is called the .

The x-coordinate of the point at which a graph crosses the x-axis is called the .

Example 4: Use Intercepts to Graph a Line

Find the x-intercept and the y-intercept of the graph of – 2𝑥 + 𝑦 – 4 = 0. Then graph the equation.

x

y

Name: Chapter 2: Linear Relations and Functions

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Lesson 2-3: Rate of Change and Slope Date:

is the ratio that compares how much one quantity changes, on

average, relative to the change in another quantity.

Real-World Example 1: Constant Rate of Change

COLLEGE ADMISSIONS In 2004, 56,878 students applied to UCLA. In 2006, 60,291 students applied.

Find the rate of change in the number of students applying for admission from 2004 to 2006.

Real-World Example 2: Average Rate of Change

BUSINESS Refer to the graph below, which shows data on the fastest-growing restaurant chain in the U.S.

during the time period of the graph. Find the rate of change of the number of stores from 2001 to 2006.

Name: Chapter 2: Linear Relations and Functions

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Example 3: Find Slope Using Coordinates

Find the slope of the line that passes through and .

Example 4: Find Slope Using a Graph

Find the slope of the line shown below.

Name: Chapter 2: Linear Relations and Functions

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Lesson 2-4: Writing Linear Equations Date:

Example 1: Write an Equation in Slope-Intercept Form

Write an equation in slope-intercept form for the line shown.

Example 2: Write an Equation Given Slope and Once Point

Write an equation of the line through (0, – 7) with a slope of 4

3 in slope-intercept form.

Name: Chapter 2: Linear Relations and Functions

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Example 3: Write an Equation Given Two Points

Write an equation of the line that passes through (2, 0) and (– 5, 8).

Example 4: Write an Equation of a Perpendicular Line

Write an equation for the line that passes through (– 2, – 1) and is perpendicular to the line whose equation

is 𝑦 = – 𝑥 + 2.

Name: Chapter 2: Linear Relations and Functions

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Lesson 2-6: Special Functions Date:

Example 1: Piecewise-Defined Function

Graph 𝑓(𝑥) = {𝑥 − 1 𝑖𝑓 𝑥 ≤ 3

−1 𝑖𝑓 𝑥 > 3. Then identify the domain and range.

Example 2: Write a Piecewise-Defined Function

Write the piecewise-defined function shown in the graph.

A. B.

x

y

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Unlike a piecewise-defined function, a function contains a single

expression. For example, a function

Real-World Example 3: Use a Step-Function

PSYCHOLOGY One psychologist charges for counseling sessions at the rate of $85 per hour or any

fraction thereof. Draw a graph that represents this situation.

𝑥 𝐶(𝑥)

0 < 𝑥 ≤ 1

1 < 𝑥 ≤ 2

2 < 𝑥 ≤ 3

3 < 𝑥 ≤ 4

4 < 𝑥 ≤ 5

Example 4: Absolute Value Functions

Graph . Identify the domain and range.

𝑥 𝑦 =

-2

-1

0

1

2

x

y

x

y

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Lesson 2-7: Parent Functions and Transformations Date:

Example 1: Identify a Function Given the Graph

A. B. C.

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Example 2: Describe and Graph Translations Example 3: Describe and Graph Reflections

Describe the translation in . Describe the reflection in .

Then graph the function. Then graph the function.

x

y

x

y

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Example 4: Describe and Graph Dilations

Describe the dilation on .

Then graph the function.

Real-World Example 5: Identify Transformation

ARCHWAYS The function 𝑓(𝑥) = −1

2(𝑥 − 5)2 + 12.5 can be used to represent a parabolic archway.

Describe the transformations in the function. Then graph the function.

x

y

x

y

Name: Chapter 2: Linear Relations and Functions

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Lesson 2-8: Graphing Linear and Absolute Value Inequalities Date:

Example 1: Dashed Boundary

Graph

Real-World Example 2: Solid Boundary

A. EDUCATION One tutoring company advertises that it specializes in helping students who have a

combined score on the SAT that is 900 or less.

Write an inequality to describe the combined scores of students who are prospective tutoring clients. Let x

represent the verbal score and y the math score. Graph the inequality.

B. Does a student with verbal score of 480 and a math score of 410 fit the tutoring company’s guidelines?

x

y

x

y

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Example 3: Absolute Value Inequality

Graph

x

y

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Lesson 2-5: Scatter Plots and Lines of Regression Date:

Example 1: Use a Scatter Plot and Prediction Equation

A. EDUCATION The table below shows the approximate percent of students who sent applications to

two colleges in various years since 1985. Make a scatter plot of the data and draw a line of fit. Describe the

correlation.

x

y

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B. Use two ordered pairs to write a prediction equation.

C. Predict the percent of students who will send applications to two colleges in 2010.

D. How accurate is this prediction?

Real-World Example 2: Regression Line

INCOME The table shows the median income of U.S. families for the period 1970–2002.

Use a graphing calculator to make a scatter plot of the data. Find an equation for and graph a line of

regression. Then use the equation to predict the median income in 2015.