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Types of functions: Let's consider the following cell phone plans
• Unlimited minutes plan– $199 per month– No extra charge for minutes
• Pay by the minute plan– $19 per month– $0.20 per minute
Minutes 100 200 300 400 500 600 700 800 900 1000 1100 1200
Unlimited Minutes $199 $199 $199 $199 $199 $199 $199 $199 $199 $199 $199 $199
Pay by the Minute $39 $59 $79 $99 $119 $139 $159 $179 $199 $219 $239 $259
0
50
100
150
200
250
300
0 500 1000 1500
Minutes
Cos
t ($
) UnlimitedMinutes
Pay by theminute
linear function
Recall slope of a line :
2 1
2 1
y ym
x x
rise = 120
run = 600
risem
run
constant function
The slope of the linear function is 120/600 = 0.2. This is reflected in the fact that the charge is $0.20 per minute.
The slope of a constant function is always zero.
or
Cost of monthly water consumption
18.9
36.4
0
10
20
30
40
50
60
70
0 5 10 15 20
Water consumed (thousands of gallons)
Cos
t ($
)
Instead of slope, we define "average rate of change" between two given x-values:
Average rate of change from 10,000 to 15,000 gallons:
36.4 18.93.5
15 10
This is a non-linear function.
$3.50 per thousand gallons
2 1
2 1
f x f x
x x
The preceding function comes from this table
City of Austin UtilityWater rates for Single Family Residential customer inside city
Monthly consumption Charge per 1000 gallons
0 to 2000 gallons $0.70
2001 to 9000 gallons $2.00
9001 to 15,000 gallons $3.50
Over 15,001 gallons $6.12
The charge per 1000 gallons can be interpreted as the average rate of change.
Note that for a non-linear function this number depends on your choice of x-values.