Warm UpEvaluate each expression for a = 2, b = –3, and c = 8. 1. a + 3c

2. ab – c

4. 4c – b

5. ba + c

26

–14

35

17

FunctionsWriting FunctionsGraphing Functions

•Identify independent and dependent variables.•Write an equation in function notation and evaluate a function for given input values.•Graph functions given a limited domain.•Graph functions given a domain of all real numbers.

Objectives

Vocabulary

independent variabledependent variablefunction rule function notation

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 and

Continued

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

10 – 4 2 and

15 – 4 3 and

20 – 4 4 and

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

Step 3 Write an equation.or The value of y is one-fifth of x.

Try This!

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

{(1, 3), (2, 6), (3, 9), (4, 12)}

x

y

1 2 3 4

3 6 9 12

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

1 3 = 3 and 1 + 2 = 3

y = 3x

Try This! Continued

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

2 • 3 = 63 • 3 = 94 • 3 = 12

2 + 2 6 3 + 2 9 4 + 2 12

The value of y is 3 times x.

Step 3 Write an equation.

The value of y is 3 times x.

The equation in Example 1 describes a function because for each x-value (input), there is only one y-value (output).

The input of a function is the independent variable. The output of a function is the dependent variable. The value of the dependent variable depends on, or is a function of, the value of the independent variable.

Identifying Independent and Dependent Variables

Identify the independent and dependent variablesin the situation.

A painter must measure a room before deciding how much paint to buy.

The amount of paint depends on the measurement of a room.

Dependent: amount of paintIndependent: measurement of the room

Identify the independent and dependent variablesin the situation.

The height of a candle decrease d centimeters for every hour it burns.

Dependent: height of candle Independent: time

The height of a candle depends on the number of hours it burns.

Identifying Independent and Dependent Variables

A veterinarian must weight an animal before determining the amount of medication.

The amount of medication depends on the weight of an animal.

Dependent: amount of medicationIndependent: weight of animal

Identify the independent and dependent variablesin the situation.

Identifying Independent and Dependent Variables

Independent – Dependent Variables

A company charges $10 per hour to rent a jackhammer.

Identify the independent and dependent variable in the situation.

The cost to rent a jackhammer depends on the length of time it is rented.

Dependent variable: cost

Independent variable: time

Identify the independent and dependent variable in the situation.

Camryn buys p pounds of apples at $0.99 per pound.

The cost of apples depends on the number of pounds bought.

Dependent variable: cost

Independent variable: pounds

Independent – Dependent Variables

Helpful Hint

There are several different ways to describe the variables of a function.

IndependentVariable

DependentVariable

x-values y-values

Domain Range

Input Output

x f(x)

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.

Writing Functions

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

Dependent: chargesIndependent: hours

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) = 40m + 100.

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 cost

Independent: number of months Let m represent the number of months

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

Steven buys lettuce that costs $1.69/lb.

The function for cost of the lettuce is f(x) = 1.69x.

The total cost depends on how many pounds of lettuce that Steven buys.

Dependent: total cost Independent: pounds

Let x represent the number of pounds Steven bought.

Writing Functions

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

An amusement park charges a $6.00 parking fee plus $29.99 per person.

The function for the total park cost is

f(x) = 29.99x + 6.

The total cost depends on the number of persons in the car, plus $6.

Dependent: total costIndependent: number of persons in the car

Let x represent the number of persons in the car.

Writing Functions

You can think of a function as an input-output machine.

input

10

x

Functionf(x)=5x

output

5x

6

30

2

Graphing a Function From a Given Equation

x (x, y)

–3 (–3, 1)

0 (0, 2)

3 (3, 3)

6 (6, 4)

Graph the function for the given domain.

D: {–3, 0, 3, 6}

Step 1 Substitute the given value of the domain for x and find values of y.

••

••

y

x

Step 2 Graph the ordered pairs.

Continued

Graph the function for the given domain.

Graph the function for the given domain.

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Graphing Functions Given a Domain

Step 1 Use the given values of the domain to find values of f(x).

f(x) = x2 – 3 (x, f(x))x

–2

–1

0

1

2

f(x) = (–2)2 – 3 = 1

f(x) = (–1)2 – 3 = –2

f(x) = 02 – 3 = –3

f(x) = 12 – 3 = –2

f(x) = 22 – 3 = 1

(–2, 1)

(–1, –2)

(0, –3)

(1, –2)

(2, 1)

••

•

•

•

y

x

Step 2 Graph the ordered pairs.

Graph the function for the given domain.

f(x) = x2 – 3; D: {–2, –1, 0, 1, 2}

Example 1B Continued

If the domain of a function is all real numbers, any number can be used as an input value. This process will produce an infinite number of ordered pairs that satisfy the function. Therefore, arrowheads are drawn at both “ends” of a smooth line or curve to represent the infinite number of ordered pairs. If a domain is not given, assume that the domain is all real numbers.

Graphing Functions Using a Domain of All Real Numbers

Step 1 Use the function to generate ordered pairs by choosing several values for x.

Step 2

Step 3

Plot enough points to see a pattern for the graph. (Usually 5 points)

Connect the points with a line or smooth curve.