Algebra 1

Chapter 4 Section 3

4-3: Writing and Graphing Functions

Objectives

Write an equation in function notation and evaluate a function for given input values.

Graph functions and determine whether an equation represents a function.

16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. Also covered: 18.0

California Standards

Suppose Tasha baby-sits and charges $5 per hour.

Time Worked (h) x 1 2 3 4

Amount Earned ($) y 5 10 15 20

The amount of money Tasha earns is $5 times the number of hours she works. You can write an equation using two variables to show this relationship.

Amount earned is $5 times the number of hours worked.

y =5 x

4-3: Writing and Graphing Functions

Example 1: Using a Table to Write an Equation

Determine a relationship between the x- and y-values. Write an equation.

x

y

5 10 15 20

1 2 3 4

Step 1 List possible relationships between the first x and y-values.

5 – 4 = 1 or

Example 1: Continued

Step 2 Determine which relationship works for the other x- and y- values.

10 – 4 2

15 – 4 3

20 – 4 4

The value of y is one-fifth, , of x.

Step 3 Write an equation.

or The value of y is one-fifth of x.

The second relationship works.

When an equation has two variables, its solutions will be all ordered pairs (x, y) that makes the equation true. Since the solutions are ordered pairs, it is possible to represent them on a graph. When you represent all solutions of an equation on a graph, you are graphing the equation.

Since the solutions of an equation that has two variables are a set of ordered pairs, they are a relation. One way to tell if this relation is a function is to graph the equation use the vertical-line test.

Example 2: Graphing Functions

Graph each equation. Then tell whether the equation represents a function.

–3x + 2 = y

Step 1 Choose several values of x and generate ordered pairs.

Step 2 Plot enough points to see a pattern.

–3(–1) + 2 = 5

–3(0) + 2 = 2

–3(1) + 2 =–1

–1

0

1

–3x + 2 = y x (x, y)

(–1, 5)

(0, 2)

(1, –1)

Example 2 Continued

Step 3 The points appear to form a line. Draw a line through all the points to show all the ordered pairs that satisfy the function. Draw arrowheads on both “ends” of the line.

Step 4 Use the vertical line test on the graph.

No vertical line will intersect the graph more than once. The equation –3x + 2 = y represents a function.

When choosing values of x, be sure to choose both positive and negative values.

Helpful Hint

Example 3: Graphing Functions

Graph each equation. Then tell whether the equation represents a function.

y = |x| + 2

Step 1 Choose several values of x and generate ordered pairs.

1 + 2 = 3

0 + 2 = 2

1 + 2 = 3

–1

0

1

|x| + 2 = y x (x, y)

(–1, 3)

(0, 2)

(1, 3)

Step 2 Plot enough points to see a pattern.

Example 3: Continued

Step 3 The points appear to form a V-shaped graph. Draw two rays from (0, 2) to show all the ordered pairs that satisfy the function. Draw arrowheads on the end of each ray.

Step 4 Use the vertical line test on the graph.

No vertical line will intersect the

graph more than once. The equation y = |x| + 2 represents a function.

Looking at the graph of a function can help you determine its domain and range.

All x-values appear somewhere on the graph.

All y-values appear somewhere on the graph.

For y = 5x the domain is all real numbers and the range is all real numbers.

y =5x

Only nonnegative y-values appear on the graph.

Looking at the graph of a function can help you determine its domain and range.

All x-values appear somewhere on the graph.

For y = x2 the domain is all real numbers and the range is y ≥ 0.

y = x2

In a function, one variable (usually denoted by x) is the independent variable and the other variable (usually y) is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable. For Tasha, who earns $5 per hour, the amount she earns depends on, or is a function of, the amount of time she works.

When an equation represents a function, you can write the equation using functional notation. If x is independent and y is dependent, the function notation for y is f(x), read “f of x,” where f names the function.

The dependent variable is a function of the independent variable.

y is a function of x.

y = f (x)

Tasha’s earnings, y = 5x, can be rewritten in function notation by substituting f(x) for y—f(x) = 5x. Note that functional notation always defines the dependent variable in terms of the independent variable.

Identify the independent and dependent variables. Write a rule in function notation for the situation.

A math tutor charges $35 per hour.

The function for the amount a math tutor charges is f(h) = 35h.

Example 4: Writing Functions

The amount a math tutor charges depends on number of hours.

Independent: timeDependent: cost

Let h represent the number of hours of tutoring.

A fitness center charges a $100 initiation fee plus $40 per month.

The function for the amount the fitness center charges is f(m) = 100 + 40m.

Example 5: Writing Functions

Identify the independent and dependent variables. Write a rule in function notation for the situation.

The total cost depends on the number of months, plus $100.

Dependent: total costIndependent: number of months

Let m represent the number of months.

You can think of a function rule as an input-output machine. For Tasha’s earnings, f(x) = 5x, if you input a value x, the output is 5x.

If Tasha wanted to know how much money she would earn by working 6 hours, she would input 6 for x and find the output. This is called evaluating the function.

Example 6: Evaluating Functions

Evaluate the function for the given input values.

For f(x) = 3x + 2, find f(x) when x = 7 and when x = –4.

= 21 + 2

f(7) = 3(7) + 2Substitute 7 for x.

f(x) = 3(x) + 2

= 23

Simplify.

f(x) = 3(x) + 2

f(–4) = 3(–4) + 2 Substitute –4 for x.

Simplify.= –12 + 2

= –10

Functions can be named with any letter; f, g, and h are the most common. You read f(6) as “f of 6,” and g(2) as “g of 2.”

Reading Math