Transcript An Introduction to Functions

1.7 An Introduction to Functions
What you should learn
GOAL
GOAL
1
Identify a function and make an input-output
table for a function.
2
Write an equation for a real-life function,
such as the relationship between water
pressure and depth.
Why you should learn it
To represent real-life relationships between two
quantities such as time and altitude for a rising hot-air
balloon.
1.6 Tables and Graphs
GOAL
1
INPUT-OUTPUT TABLES
VOCABULARY
•function
•input/domain
•output/range
In a function, each input has
exactly one output. Another way
to put it is no number in the
input can be repeated.
•input-output table
EXAMPLE 1
Extra Example 1
The profit on the school play is $4 per ticket minus $280,
the expense to build the set. There are 300 seats in the
theater. The profit for n tickets sold is
p = 4n – 280 for 70 ≤ n ≤ 300.
a. Make an input-output table.
n
70
71
72
73
… 300
p
0
4
8
12
… 920
b. Is this a function? Yes; none of the inputs are repeated.
c. Describe the domain and range.
Domain: 70, 71, 72, 73,… , 300
Range: 0, 4, 8, 12,… ,920
EXAMPLE 2
Extra Example 2
You bicycle 4 mi and decide to ride for 2.5 more hours at 6
mi/hr. The distance you have traveled d after t hours is
given by d = 4 + 6t, where 0 ≤ t ≤ 2.5.
a. Make an input-output table. Calculate d for each half-hour
(t = 0, 0.5, 1, 1.5, 2, 2.5).
t
0
0.5
1
1.5
2
2.5
d
4
7
10
13
16
19
b. Draw a line graph.
Extra Example 2 (cont.)
t
0
0.5
1
1.5
2
2.5
d
4
7
10
13
16
19
Bicycle Distance
Distance (miles)
20
15
10
5
0
0
0.5
1
1.5
Time (hours)
2
2.5
4 WAYS TO DESCRIBE A FUNCTION
• Input-Output Table
• Description in Words
• Equation
• Graph
By the end of the lesson you should be able to move
comfortably among all four representations. You will then
have a variety of ways to model real-life situations.
Checkpoint
A plane is at 2000 ft. It climbs at a rate of 1000 ft/min for
4 min. The altitude h for t minutes is given by
h = 2000 + 1000t for 0 ≤ t ≤ 4.
1. Make a table (use 0, 1, 2, 3, and 4 minutes).
2. Draw a line graph.
3. Describe the domain and range.
Checkpoint (cont.)
t
0
d
1
2
3
4
2000 3000 4000 5000
Plane Altitude
Height (feet)
8000
6000
4000
2000
0
0
1
2
3
Time (minutes)
4
6000
Checkpoint (cont.)
0
1
2
3
4
d 2000 3000 4000 5000 6000
Domain: all numbers between
and including 0 and 4
Range: all numbers between
and including 2000 and 6000
8000
Height (feet)
t
Plane Altitude
6000
4000
2000
0
0
1
2
3
Time (minutes)
All numbers are included
because time is continuous.
This is what is shown by
connecting the data points with
a line. Even numbers such as
1.73 minutes or 2148.4 ft are
included as the plane climbs.
4
1.7 An Introduction to Functions
GOAL
2
WRITING EQUATIONS FOR FUNCTIONS
Use the problem solving strategy from Section 1.5 to:
•Write a verbal model
•Assign labels
•Write an algebraic model
EXAMPLE 3
Extra Example 3
An internet service provider charges $9.00 for the first 10
hours and $0.95 per hour for any hours above 10 hours.
Represent the cost c as a function of the number of hours
(over 10) h.
a. Write an equation.
b. Create an input-output table for hours 10-14.
c. Make a line graph.
Extra Example 3 (cont.)
VERBAL
MODEL
Cost
LABELS
c
=
$9
ALGEBRAIC
MODEL
h
c
Connection
+
fee
Rate
per
hour
• Number
of hours
$0.95
c = $9 + $0.95h
10
9
11
9.95
12
13
14
10.90 11.85 12.80
h
Extra Example 3 (cont.)
h
c
10
9
11
9.95
12
13
14
10.90 11.85 12.80
Internet Cost
14
Amount ($)
12
10
8
6
4
2
0
10
11
12
Time (hours)
13
14
Checkpoint
The temperature at 6:00 a.m. was 62°F and rose 3°F
every hour until 9:00 a.m. Represent the temperature
T as a function of the number of hours h after 6:00 a.m.
1. Write an equation.
2. Make an input-output table, using a one-half hour
interval.
3. Make a line graph.
Checkpoint (cont.)
a. T = 62 + 3h
b.
h
0
0.5
1
1.5
2
2.5
3
T
62
63.5
65
66.5
68
69.5
71
Temperature (F)
Temperature Change
c.
75
70
65
60
55
0
0.5
1
1.5
Time (hours)
2
2.5
3
QUESTIONS?