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.