1. Algebra 1
Chapter 4 Section 3

2. 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.

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. y x =5 4-3: Writing and Graphing Functions
Suppose Tasha baby-sits and charges $5 per hour.
Amount Earned ($) y 5 10 15 20
Time Worked (h) x 1 2 3 4
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

5. 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

6.       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.

7. 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.

9. Example 2: Graphing Functions
Graph each equation. Then tell whether the equation represents a function.
Step 2 Plot enough points to see a pattern.
–3x + 2 = y
Step 1 Choose several values of x and generate ordered pairs. 
–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)

10. 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. 

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

12. Example 3: Graphing Functions
Graph each equation. Then tell whether the equation represents a function.
Step 2 Plot enough points to see a pattern.
y = |x| + 2
Step 1 Choose several values of x and generate ordered pairs. 
1 + 2 = 3
0 + 2 = 2 –1 1
|x| + 2 = y x
(x, y)
(–1, 3)
(0, 2)
(1, 3)

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. Example 4: 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.

19. Example 5: 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.

20. 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.

21. 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.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute
7 for x.
f(–4) = 3(–4) + 2
Substitute
–4 for x.
f(7) = 3(7) + 2 =
Simplify.
Simplify.
= –12 + 2
= 23
= –10

22. 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