1. Preview
Warm Up
California Standards
Lesson Presentation

2. Warm Up Evaluate each expression for a = 2, b = –3, and c = 8.
2. ab – c 3. 26
–14 1 2
c + b 1
4. 4c – b 35
5. ba + c 17
Solve each equation for y.
6. 2x + y = 3
y = –2x + 3
7. –x + 3y = –6
8. 4x – 2y = 8
y = 2x – 4

3. California
Standards
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

4. Vocabulary
dependent variable
independent variable
function notation

5. y x =5 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

6. Additional 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

7. Additional 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 second relationship works.
The value of y is one-fifth, , of x.
Step 3 Write an equation. or
The value of y is one-fifth of x.

8. Check It Out! Example 1
Determine a relationship between the x- and y-values. Write an equation.
{(1, 3), (2, 6), (3, 9), (4, 12)} x 1 2 3 4 y 3 6 9 12
Step 1 List possible relationships between the first x- and y-values.
1  3 = 3 or = 3

9. Check It Out! Example 1 Continued
Step 2 Determine which relationship works for the other x- and y- values.  
2 • 3 = 6
3 • 3 = 9
4 • 3 = 12
2 + 2  6
3 + 2  9
4 + 2  12    
The first relationship works. The value of y is 3 times x.
Step 3 Write an equation.
y = 3x
The value of y is 3 times x.

10. 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.

12. Additional Example 2A: 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 1
–3x + 2 = y x
(x, y)
(–1, 5)
(0, 2)
(1, –1)

13. Additional Example 2A 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. 

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

15. Additional Example 2B: 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.
Step 2 Plot enough points to see a pattern.
1 + 2 = 3
0 + 2 = 2 –1 1
|x| + 2 = y x
(x, y)
(–1, 3)
(0, 2)
(1, 3) 

16. Additional Example 2B 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.

17. Step 2 Plot enough points to see a pattern.
Check It Out! Example 2a
Graph each equation. Then tell whether the equation represents a function.
y = 3x – 2
Step 2 Plot enough points to see a pattern.
Step 1 Choose several values of x and generate ordered pairs. 
3(–1) – 2 = –5 –1 1
3x – 2 = y x
(x, y)
(–1, –5)
(0, –2)
(1, 1)
3(0) – 2 = –2
3(1) – 2 = 1

18. Check It Out! Example 2a 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 y = 3x – 2 represents a function.

19. Step 2 Plot enough points to see a pattern.
Check It Out! Example 2b
Graph each equation. Then tell whether the equation represents a function.
y = |x – 1|
Step 2 Plot enough points to see a pattern.
Step 1 Choose several values of x and generate ordered pairs.
(2, 1)
(–1, 2)
(0, 1)
(1, 0)
(x, y)
y = |x – 1| x
2 = |–1 – 1|
1 = |0 – 1|
0 = |1 – 1|
1 = |2 – 1| –1 1 2 

20. Check It Out! Example 2b Continued
Step 3 The points appear to form a V-shaped graph. Draw two rays from (1, 0) 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 – 1| represents a function. 

21. Looking at the graph of a function can help you determine its domain and range.
All y-values appear somewhere on the graph.
y =5x
All x-values appear somewhere on the graph.
For y = 5x the domain is all real numbers and the range is all real numbers.

22. For y = x2 the domain is all real numbers and the range is y ≥ 0.
Looking at the graph of a function can help you determine its domain and range.
Only nonnegative y-values appear on the graph.
y = x2
All x-values appear somewhere on the graph.
For y = x2 the domain is all real numbers and the range is y ≥ 0.

23. 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.

24. 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 x. = 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.

25. Additional Example 3A: Writing Functions
Identify the independent and dependent variables. Write a rule in function notation for the situation.
A math tutor charges $35 per hour.
The amount a math tutor charges depends on number of hours.
Independent: time
Dependent: cost
Let h represent the number of hours of tutoring.
The function for the amount a math tutor charges is f(h) = 35h.

26. Additional Example 3B: Writing Functions
Identify the independent and dependent variables. Write a rule in function notation for the situation.
A fitness center charges a $100 initiation fee plus $40 per month.
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.
The function for the amount the fitness center charges is f(m) = m.

27. Check It Out! Example 3a
Identify the independent and dependent variables. Write a rule in function notation for the situation.
A tutor’s fee for music lessons is $28 per hour for private lessons.
The total cost depends on how many hours of lessons that are given.
Dependent: total cost
Independent: lessons given
Let x represent the number of lessons given.
The function for cost of music lessons is f(x) = 28x.

28. Check It Out! Example 3b
Identify the independent and dependent variables. Write a rule in function notation for the situation.
Steven buys lettuce that costs $1.69/lb.
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.
The function for cost of the lettuce is f(x) = 1.69x.

29. Check It Out! Example 3c
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 total cost depends on the number of persons in the car, plus $6.
Dependent: total cost
Independent: number of persons in the car
Let x represent the number of persons in the car.
The function for the total park cost is
f(x) = x.

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

31. Additional Example 4A: 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.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute
7 for x.
Substitute
–4 for x.
f(7) = 3(7) + 2
f(–4) = 3(–4) + 2 =
Simplify.
Simplify.
= –12 + 2
= 23
= –10

32. Additional Example 4B: Evaluating Functions
Evaluate the function for the given input values.
For g(t) = 1.5t – 5, find g(t) when t = 6 and when t = –2.
g(t) = 1.5t – 5
g(t) = 1.5t – 5
g(6) = 1.5(6) – 5
g(–2) = 1.5(–2) – 5
= 9 – 5
= –3 – 5
= 4
= –8

33. Additional Example 4C: Evaluating Functions
Evaluate the function for the given input values.
For , find h(r) when r = 600 and when r = –12.
= 202
= –2

34. 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

35. Check It Out! Example 4
Evaluate the function for the given input values.
For h(c) = 2c – 1, find h(c) when c = 1 and when c = –3.
h(c) = 2c – 1
h(c) = 2c – 1
h(1) = 2(1) – 1
h(–3) = 2(–3) – 1
= 2 – 1
= –6 – 1
= 1
= –7

36. Lesson Quiz: Part I
1. Graph y = |x + 3|.

37. Lesson Quiz: Part Il
Identify the independent and dependent variables. Write a rule in function notation for each situation.
2. A buffet charges $8.95 per person.
independent: number of people
dependent: cost
f(p) = 8.95p
3. A moving company charges $130 for weekly truck rental plus $1.50 per mile.
independent: miles
dependent: cost
f(m) = m

38. Lesson Quiz: Part III
Evaluate each function for the given input values.
4. For g(t) = find g(t) when t = 20 and when t = –12.
g(20) = 2
g(–12) = –6
5. For f(x) = 6x – 1, find f(x) when x = 3.5 and when x = –5.
f(3.5) = 20
f(–5) = –31