Transcript Algebra 1 - Teacher Pages

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.
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
4-3: 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
Example 1: Using a Table to Write an Equation
Determine a relationship between the x- and
y-values. Write an equation.
x
5
y
1
10 15 20
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 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.
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.
x
–3x + 2 = y
(x, y)
Step 2 Plot enough
points to see a pattern.


–1 –3(–1) + 2 = 5 (–1, 5)
0
–3(0) + 2 = 2
(0, 2)
1
–3(1) + 2 =–1 (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.
Helpful Hint
When choosing values of x, be sure to choose
both positive and negative values.
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.
x
|x| + 2 = y
(x, y)
–1
1+2=3
(–1, 3)
0
0+2=2
(0, 2)
1
1+2=3
(1, 3)



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.
y =5x
All y-values
appear
somewhere
on the
graph.
All x-values appear
somewhere on the graph.
For y = 5x the domain is all real numbers and the
range is all real numbers.
Looking at the graph of a function can help you
determine its domain and range.
y = x2
Only
nonnegative
y-values
appear on
the graph.
All x-values appear
somewhere on the graph.
For y = x2 the domain is all real numbers and the
range is y ≥ 0.
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
y
y
is
=
a function of
a function of
f
the independent variable.
x.
(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.
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.
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) = 100 + 40m.
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.
f(x) = 3(x) + 2
f(x) = 3(x) + 2
Substitute f(–4) = 3(–4) + 2 Substitute
f(7) = 3(7) + 2 7 for x.
–4 for x.
Simplify.
Simplify.
= 21 + 2
= –12 + 2
= 23
= –10
Reading Math
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.”