Chapter 2

Linear Functions and Relations

In Chapter 2, You Will… Move from simplifying variable expressions

and solving one-step equations and inequalities to working with two variable equations and inequalities.

Learn how to represent function relationships by writing and graphing linear equations and inequalities.

By graphing data and trend lines, you will understand how the slope of a line can be interpreted in real-world situations.

2-1 Relations and Functions

What You’ll Learn …

To graph relations. To identify functions.

Example 1 Graphing a Relation

[(-2,4), (3,-2), (-1,0), (1,5)] [(0,4),(-2,3),(-1,3),(-2,2),(1,-3)]

A relation is a set of pairs of input and output numbers.

Finding Domain and Range

The domain of a relation is the set of all inputs, or x-coordinates of the ordered pairs.

The range of a relation is the set of all outputs, or y-coordinates of the ordered pairs.

(2,4),(3,4.5),(4,7.5),(5,7),(6,5),(6,7.5)

D= _____________R= _____________

D= _____________

R= _____________

Using a Mapping

-2

-1

6

-1

0

1

3

Another way to show a relation is to use a mapping diagram, which links elements of the domain with corresponding elements of the range.

(-2,-1) (-1,-1) (-2,1) (6,3)

Example 3 Making a Mapping Diagram

(0,2) (1,3) (2,4) (2,8) (-1,5) (0,8) (-1,3) (-2,3)

A function is a relation that assigns exactly one value in the range to each value in the domain

X Y

-4 -4

-1 -4

0 -4

3 -4

X Y

1 -3

6 -2

9 -1

1 3

Example 4 Identifying Functions

-2

0

5

-1

3

4

-1

3

5

-1

0

2

3

One way you can tell whether a relation is a function is to analyze the graph of the relation using the vertical line test. If any vertical line passes through more than one point of the graph, the relation is NOT a function.

Which are Functions?

•

•• •

• •

Another way to write a function

y = 3x + 4 is f(x)= 3x + 4.

You read f(x) as “f of x” or “f is a function of x”.

The notations g(x) and h(x) also indicate functions of x.

Function Notation

Evaluating Functions

A function rule is an equation that describes a function. You can think of a function rule as an input-output machine

Function RuleFunction RuleInput Output

3x + 4

Function Rule y = 3x +

4

Evaluating a Function

Input Output

X Y

1

2

3

x y

Evaluating a Function Rule

f(n)= -3n – 10

Find f(6).

g(x) = -2x² + 7

Find g(6).

Example 6 Real World Connection

The area of a square tile is a function of the length of a side of the square. Write a function rule for the area of a square. Evaluate the function for a square tile with side length 3.5 in.

7-6 Function Operations

2.01 Use the composition of functions to model and solve problems; justify results.

What you’ll learn … To add, subtract,

multiply and divide functions

To find the composite of two numbers

Function Operations

Addition (f+g)(x) = f(x)+g(x)

Multiplication (fg)(x) = f(x) g(x)

Subtraction (f-g)(x) = f(x) – g(x)

Division (x)= , g(x)≠0

fg

f(x)g(x)

Example 1 Adding and Subtracting Functions Let f(x) = 3x +8 and g(x) = 2x-12. Find f+g and f - g and their domain.

Let f(x) = 5x2 - 4x and g(x) = 5x+1. Find f+g and f - g and their domain.

Example 2 Multiplying and Dividing Functions Let f(x) = x2 - 1 and g(x) = x+1. Find fg and and their domain.

Let f(x) = 6x2 +7x - 5 and g(x) = 2x-1. Find fg and and their domain.

fg

fg

Composition of Functions

The composition of function g with function f iswritten as g°f and is defined as (g°f)(x)= g(f(x)), where the domain of g°f consists of the values a in

the domain of f such that f(a) is in the domain of g.

(g°f)(x) = g( f(x) ) 2. Then use your answer as

the input of the outer function g(x).

1. Evaluate the inner function f(x) first.

Example 3 Composition of Functions

Let f(x) = x-2 and g(x) = x2. Find (g°f)(-5).

Let f(x) = x-2 and g(x) = x2. Find (f°g)(x) and evaluate (f°g)(-5).

Example 4a Real World Connection

Suppose you are shopping in the store in the photo. You have a coupon worth $5 off any item.

a. Use functions to model discounting an item by 20% and to model applying the coupon.

b. Use a composition of your two functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon.

c. Use a composition of your two functions to model how much you would pay for an item if the clerk applies the coupon first and then the discount.

d. How much more is any item if the clerk applies the coupon first?

Example 4b Real World Connection A store is offering a 10% discount on all items. In

addition, employees get a 25% discount.

a. Write a composite function to model taking the 10% discount first.

b. Write a composite function to model taking the 25% discount first.

c. Suppose you are an employee. Which discount would you prefer to take first?

7-7 Inverse Relations and Functions

2.01 Use the composition of functions to model and solve problems; justify results.

What you’ll learn … To find the inverse

of a relation or function.

The Inverse of a Function

If a relation maps element a of its domain to element b of its range., the inverse relation “undoes” the relation and maps b back to a.

Relation r Inverse of r

1.21.41.61.9

1

2

1.21.41.61.9

1

2

Example 1 Finding the Inverse of a Relation

Find the inverse of relation s.

Graph s and its inverse.

x -1 0 1 1

y 1 2 3 4

Example 2 Interchanging x and y

Find the inverse of y = x2 + 3.

Does y = x2 + 3 define a function? Is its inverse a function? Explain.

Find the inverse of y = 3x - 10.

Is its inverse a

function? Explain.

Example 3 Graphing a Relation and Its Inverse

Graph y= x2 + 3 and its inverse, y = +√x -3 .

Graph y= 3x-10 and its inverse.

The inverse of a function is denoted by f-1. Read f-1 as “the inverse of f” or as “f inverse”. The notation f(x) is used for functions, but f-1(x) may be a relation that is not a function.

Example 4a Finding an Inverse Function

Consider the function f(x) = √x+1.

a. Find the domain and range of f.

b. Find f-1.

c. Find the domain and range of f-1.

d. Is f-1 a function? Explain.

Example 4b Finding an Inverse Function

Consider the function f(x) = 10 – 3x.

a. Find the domain and range of f.

b. Find f-1.

c. Find the domain and range of f-1.

d. Find f-1(f(3)).

e. Find f-1(f(2)).

Composite of Inverse Functions

If f and f-1 are inverse functions then, (f-1°f)(x) and (f°f-1)(x) = x.

Example 6 Composite of Inverse Functions

For f(x) = 5x + 11, find (f-1°f)(777).

For f(x) = 5x + 11, find (f°f-1)(-5802).

2-2 Linear Equations

What you’ll learn … To graph linear equations. To write equations of lines.

Graphing Linear Equations

A function whose graph is a line is a linear function. You can represent a linear function with a linear equation, such as y=3x+2. A solution is any ordered pair (x,y) that makes the equation true.

Because the y depends on the value of x, y is called the dependent variable and the x is called the independent variable.

Example 1 Graphing a Linear Equation

Graph the equations using a table.

y=-3x

y=½x+3

x y x y

The y intercept of a line is the point in which the line crosses the y-axis.

The x intercept of a line is the point in which the line crosses the x-axis.

The standard form of a linear equation is Ax +By = C, where A,B and C are real numbers and A and B are not both zero.

Example 2 Real World Connection

The equation 3x +2y =120 models the number of passengers who can sit in a train car, where x is the number of adults and y is the number of children. Graph the equation. Describe the domain and range. Explain what the x and y intercepts represent.

Slope The slope of a non-vertical line is the ratio

of the vertical change to a corresponding horizontal change.

You can calculate the slope by subtracting the corresponding coordinates of two points on the line.

Slope Formula

Vertical change (rise) Horizontal change (run)

y2 – y1 x2 – x1

=

Example 3 Finding Slope

Find the slope of the line through the points(3,2) and (-9,6).

Find the slope of the line through the points (5,2) and (-6,2).

Point-Slope Form

When you know the slope and a point on a line, you can use the point-slope form to write an equation of the line.

y – y1 = m (x - x1)

Example 4 Writing an Equation Given the Slope and a Point

Write in standard form an equation of the line with slope -½ through the point (8,-1).

Write in standard form an equation of the line with slope 2 through the point (4, -2).

Example 5 Writing an Equation Given Two Points

Write in point slope form an equation of the line through (1,5) and (4,-1).

Write in point slope form an equation of the line through (-2,-1) and (-10,17).

Slope Intercept Form

Another form of the equation of a line is slope intercept form, which you can use to find the slope by examining the equation.

y= mx +b

Slope y intercept

Example 6 Finding Slope Using Slope-Intercept Form

Find the slope of 4x + 3y = 7.

Find the slope of ½x + ¾y = 1

Summary: Equations of Lines

Point SlopeForm

y-y1 = m(x-x1) y- 2= -3(x+4)

Standard Form Ax + By = C 3x + y = -10

Slope Intercept Form

y = mx + b y = -3x - 10

Special Slopes

Horizontal Line Vertical Line

Zero Slope Undefined Slope

Special Slopes

Perpendicular Lines Parallel Lines

Have reciprocal slopes Have same slopes

Example 7 Writing an Equation of a Perpendicular or Parallel Line

Write an equation perpendicular to y=5x-3

And through the point (-1,3).

Write an equation parallel to y=2/3x+5/8

and through the point (2,1).

2-4 Using Linear Models

2.04 Create and use best-fit mathematical models of linear, exponential, and quadratic functions to solve problems involving sets of data.

a) Interpret the constants, coefficients, and bases in the context of the data.

b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

What you’ll learn … To write linear

equations that model real-world data.

To make predictions from linear models.

Example 1a Real World Connection

Jacksonville, Florida has anelevation of 12 ft above sea level. A hot air balloon taking off from Jacksonville rises 50 ft/min. Write anequation to model the balloon’s elevation as a function of time.

Graph the equation.

Example 1b Real World Connection Suppose a balloon begins

descending at a rate of 20 ft/min from an elevation of 1350 ft. Write an equation to model the balloon’s elevation as a function of time.

What is true about the slope of this line?

Graph the equation. Interpret the h-intercept.

Example 2a Real World Connection A candle is 6 inches tall after burning for 1hour. After 3 hours, it is 5½ inches tall. Write a linear equation to model theheight y of the candle after burning x hours.

What does the slope represent ? The y intercept?

Graph the equation.

Example 2b Real World Connection

Another candle is 7inches tall after burning for 1hour. After 2 hours, it is 5inches tall. Write a linearequation to model the height y of the candle after burning x hours.

How tall will the candle be after burning 11 hours? What was the original height of the candle? When will the candle

burn out?

Scatter Plots A scatter plot is a graph that relates

two different sets of data by plotting the data as ordered pairs. You can use a scatter plot to determine a relationship between the data sets.

A trend line is a line that approximates the relationship between the data sets of a scatter plot. You can use a trend line to make predictions.

Correlation

Strong, Positive Correlation

Strong, Negative Correlation

No Correlation

Weak, Positive Correlation

Weak, Negative Correlation

Calculator Steps Enter data into lists.

• Stat Edit Turn on Stat Plot.

• 2nd y = Zoom 9 Turn on Diagnostic

• 2nd Zero Find the trend line.

• Stat Calc 4

Trend Line

Fat Calories

6 267

7 260

10 220

19 388

20 430

27 550

36 633

Make a scatter plot of the data.

Draw a trend line.

Estimate the number of calories in a fast-food item that has 14g of fat.

Trend Line

Hawk Length Wingspan

Cooper’s 21 36

Crane 21 41

Gray 18 38

Harris’s 24 46

Roadside 16 31

Broad-winged 19 39

Short-tailed 17 35

Swanson’s’ 19 46

Make a scatter plot of the data on the calculator.

Draw a trend line.

Predict the wingspan of a hawk that is 28 inches long.

2-5 Absolute Value Functions and Graphs

2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

What you’ll learn … To graph absolute

value functions.

Graphing Absolute Value Functions A function of the form f(x) = mx+b +c,

where m≠0, is an absolute value function.

The vertex of a function is a point where the function reaches a maximum or minimum.

Example 1 Graphing an Absolute Value Function

Graph y= 3x+12 Graph y= - x +1 - 2

Example 2 Using a Graphing Calculator

Graph y= 3 - ½x

Vertex _________

Graph y= - 3x+4 +6

Vertex __________

Example 4 Real World Connection

Suppose you pass the Betsy Ross House halfway along your trip to school each morning. You walk at a rate of one city block per minute. Sketch a graph of your trip to school based on your distance and time from the Betsy Ross House.

Blo

cks

from

Ross

House

Minutes before arrival Minutes after departure

Example 4 Real World Connection

Suppose you ride your bicycle to school at a rate of three city blocks per minute. How would the graph of your trip to school change? Sketch a new graph.

Blo

cks

from

Ross

House

Minutes before arrival Minutes after departure

2-6 Vertical and Horizontal Translations

2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

What you’ll learn … To analyze vertical

translations. To analyze

horizontal translations.

Translating Graphs Vertically

A translation is an operation that shifts a graph horizontally, vertically or both. It results in a graph of the same slope and size, in a different position.

Vertical Translation Horizontal Translation

Example 1 Comparing Graphs

Compare the graphs y = x and y = x -3

Compare the graphs y = x and y = x +5

Rules of Translations

Given f(x), then –f(x) is a reflection about the x axis.

Given f(x), then f(x) + k moves up k units.

Given f(x), then f(x) - k moves down k units.

Given f(x), then f(x+k) moves left k units.

Given f(x), then f(x-k) moves right k units.

A family of functions is a group of functions with common characteristics. A parent function is the simplest function with these characteristics. A parent function and one or more translations make up a family of functions.

Example 2 Graphing a Vertical Translation

For each function, identify the parent function and the value of k. Then graph the function by translating the parent function.a) y = x – 1b) y = 3x + 5c) y = x – 3d) y = - x +2e) y = x +1 +2f) y = - x - 1 -3

Example 3a Writing Equations for Vertical Translations

Write an equation for each translation.

a) y = 2x, 4 units down.

b) y = 3x , 2 units down.

Example 3b Writing Equations for Vertical Translations

Write an equation for each translation given y = x .

Example 4 Graphing a Horizontal Translation

For each function, identify the parent function and the value of k. Then graph the function by translating the parent function.a) y = x + 3 b) y = - x - 2c) y = x - 1 d) y = - x + ¾

Example 5 Writing Equations for Horizontal Translations

Write an equation for each translation given y = x .

Example 6 Real World Connection Describe a possible translation of Figures A and

B in the Nigerian textile design below.

AB

2-7 Two- Variable Inequalities

2.08 Use equations and inequalities with absolute value to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

What you’ll learn … To graph linear

inequalities. To graph absolute

value inequlaities.

Graphing Linear Inequalities

A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.

To graph a linear inequality, first graph the boundary line.

Then decide which side of the line contains solutions to the inequality and whether the boundary line is included.

Example 1a Graphing an Inequality

y > 2x + 3

m= ____b = ____

Example 1b Graphing an Inequality

3x - 5y ≥ 10

m= ____b = ____

Example 2 Real World Connection

At least 35 performers of the Big Tent Circus are in the grand finale. Some pile into cars, while others balance on bicycles. Seven performers are in each car, and five performers are on each bicycle. Draw a graph showing allthe possible combination of cars and bicycles that could be use in the finale.

Example 3 Graphing Absolute Value Inequalities

y ≤ x – 4 + 5 -y + 3 > x + 1

Example 4 Writing Inequalities

Write an inequality for each graph. The boundary line is given.

In Chapter 2, You Should Have Moved from simplifying variable

expressions and solving one-step equations and inequalities to working with two variable equations and inequalities.

Learned how to represent function relationships by writing and graphing linear equations and inequalities.

By graphing data and trend lines, you should understand how the slope of a line can be interpreted in real-world situations.