Transcript 4-3 Writing and Graphing Functions

4-3 Writing and Graphing Functions
Preview
Warm Up
California Standards
Lesson Presentation
4-3 Writing and Graphing Functions
Warm Up
Evaluate each expression for a = 2, b = –3,
and c = 8.
1. a + 3c 26
2. ab – c –14
3. 1
c+b 1
2
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
4-3 Writing and Graphing Functions
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
Vocabulary
dependent variable
independent variable
function notation
4-3 Writing and Graphing Functions
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
Additional 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
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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 1 + 2 = 3
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
4-3 Writing and Graphing Functions
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.
x
–3x + 2 = y
Step 2 Plot enough
points to see a pattern.

(x, y)
–1 –3(–1) + 2 = 5 (–1, 5)
0
–3(0) + 2 = 2
(0, 2)
1
–3(1) + 2 =–1 (1, –1)


4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
Helpful Hint
When choosing values of x, be sure to choose
both positive and negative values.
4-3 Writing and Graphing Functions
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.
x
|x| + 2 = y
Step 2 Plot enough
points to see a pattern.
(x, y)
–1
1+2=3
(–1, 3)
0
0+2=2
(0, 2)
1
1+2=3
(1, 3)



4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
Check It Out! Example 2a
Graph each equation. Then tell whether the
equation represents a function.
y = 3x – 2
Step 1 Choose several
values of x and generate
ordered pairs.
x
3x – 2 = y
Step 2 Plot enough
points to see a pattern.

(x, y)
–1 3(–1) – 2 = –5 (–1, –5)
0
3(0) – 2 = –2
(0, –2)
1
3(1) – 2 = 1
(1, 1)


4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
Check It Out! Example 2b
Graph each equation. Then tell whether the
equation represents a function.
y = |x – 1|
Step 2 Plot enough
Step 1 Choose several
points to see a pattern.
values of x and generate
ordered pairs.
x
y = |x – 1|
(x, y)
–1
2 = |–1 – 1|
(–1, 2)
0
1 = |0 – 1|
(0, 1)
1
0 = |1 – 1|
(1, 0)
2
1 = |2 – 1|
(2, 1)




4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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
the independent variable.
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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) = 100 + 40m.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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) = 6 + 29.99x.
4-3 Writing and Graphing Functions
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.
4-3 Writing and Graphing Functions
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
f(7) = 3(7) + 2 Substitute f(–4) = 3(–4) + 2 Substitute
7 for x.
–4 for x.
Simplify.
= –12 + 2
= 21 + 2
Simplify.
= 23
= –10
4-3 Writing and Graphing Functions
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
4-3 Writing and Graphing Functions
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
4-3 Writing and Graphing Functions
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.”
4-3 Writing and Graphing Functions
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(1) = 2(1) – 1
h(c) = 2c – 1
h(–3) = 2(–3) – 1
=2–1
= –6 – 1
=1
= –7
4-3 Writing and Graphing Functions
Lesson Quiz: Part I
1. Graph y = |x + 3|.
4-3 Writing and Graphing Functions
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) = 130 + 1.50m
4-3 Writing and Graphing Functions
Lesson Quiz: Part III
Evaluate each function for the given input
values.
4. For g(t) =
t = –12.
g(20) = 2
g(–12) = –6
find g(t) when t = 20 and when
5. For f(x) = 6x – 1, find f(x) when x = 3.5 and when
x = –5.
f(3.5) = 20
f(–5) = –31