1.7 Represent Functions as Graphs

Vocabulary

• Relation — an pairing of a set of inputs with a set out outputs (includes functions)

Graph a function

EXAMPLE 1

STEP 1 Make an input-output table.

SOLUTION

Graph the function y = x with domain 0, 2, 4, 6, and 8. 12

x 0 2 4 6 8

y 0 1 2 3 4

Graph a function

EXAMPLE 1

STEP 2

Plot a point for each ordered pair (x, y).

GUIDED PRACTICE for Example 1

1. Graph the function y = 2x - 1 with domain 1, 2, 3, 4, and 5.

STEP 1 Make an input-output table.

SOLUTION

x 1 2 3 4 5

y 1 3 5 7 9

STEP 2

Plot a point for each ordered pair (x, y).

Graph a function

EXAMPLE 2

The table shows the average scores on the mathematics section of the Scholastic Aptitude Test (SAT) in the United States from 1997 to 2003 as a function of the time t in years since 1997. In the table, 0 corresponds to the year 1997, 1 corresponds to 1998, and so on. Graph the function.

519516514514511512511Average score, s

6543210Years since 1997, t

Sat Scores

SOLUTION

STEP 1 Choose a scale. The scale should allow you to plot all the points on a graph that is a reasonable size.

EXAMPLE 2 Graph a function

The s-values range from 511 to 519, so label the s-axis from 5 – 10 to 520 in increments of 2 units.

The t-values range from 0 to 6, so label the t-axis from 0 to 6 in increments of 1 unit.

STEP 2

Plot the points

EXAMPLE 2 Graph a function

EXAMPLE 2GUIDED PRACTICE for Example 2

WHAT IF? In Example 2, suppose that you use a scale on the s-axis from 0 to 520 in increments of 1 unit. Describe the appearance of the graph.

2.

The graph would be very large with all the points near the top of the graph.

ANSWER

Write a function rule for a graph

EXAMPLE 3

Write a rule for the function represented by the graph.Identify the domain and the range of the function.

SOLUTION

STEP 1 Make a table for the graph.

x 1 2 3 4 5

y 2 3 4 5 6

Write a function rule for a graph

EXAMPLE 3

STEP 2 Find a relationship between the inputs and the outputs. Notice from the table that each output value is 1 more than the corresponding input value.

STEP 3

Write a function rule that describes the relationship:y = x + 1.

A rule for the function is y = x + 1 . The domain of the function is 1, 2, 3, 4, and 5. The range is 2, 3, 4, 5, and 6.

ANSWER

GUIDED PRACTICE for Example 3

Write a rule for the function represented by the graph. Identify the domain and the range of the function.

3.

ANSWER y = 5 – x; domain: 0, 1, 2, 3, and 4, range: 1, 2, 3, 4, and 5

GUIDED PRACTICE for Example 3

Write a rule for the function represented by the graph. Identify the domain and the range of the function.

ANSWER y = 5x + 5; domain: 1, 2, 3, and 4, range: 10, 15, 20 and 25

4.

Analyze a graph

EXAMPLE 4

The graph shows guitar sales (in millions of dollars) for a chain of music stores for the period 1999–2005. Identify the independent variable and the dependent variable. Describe how sales changed over the period and how you would expect sales in 2006 to compare to sales in 2005.

Guitar Sales

Analyze a graph

EXAMPLE 4

SOLUTION

The independent variable is the number of years since1999. The dependent variable is the sales (in millions of dollars). The graph shows that sales were increasing. If the trend continued, sales would be greater in 2006 than in 2005.

EXAMPLE 4GUIDED PRACTICE for Example 4

Yes; the graph seems to increase about $0.2 million every two years.

ANSWER

Based on the graph in Example 4, is $1.4 million a reasonable prediction of the chain’s sales for 2006? Explain.

5. REASONING