Chapter 2 – Linear and Exponential Functions

2.1 – Introducing Linear Models

2.2 – Introducing Exponential Models

2.3 – Linear Model Upgrades

A linear function models any process that has a constant rate of change.

m = change in y-value

change in x-value

The graph of a linear function is a straight line.

A linear function has the form:

y = f(x) = b + mx

where

f is the name of the function.

b is the starting value or y intercept (f(0)).

m is the constant rate of change or slope.

slope intercept form

2.1

In summer of 2001, the exchange rate for the Mexican peso was 9.2.

xx(dollar)(dollar)

00 11 22 33 1010

yy(peso)(peso)

00 9.29.2 18.418.4 27.627.6 9292

xx 0 to 10 to 1 1 to 21 to 2 0 to 30 to 3 1 to 101 to 10

change in xchange in x 11 11 33 99

yy 0 to 9.20 to 9.2 9.2 to 18.49.2 to 18.4 0 to 27.60 to 27.6 9.2 to 929.2 to 92

change in ychange in y 9.29.2 9.29.2 27.627.6 82.882.8

mm 9.2/19.2/1

9.29.2

9.2/19.2/1

9.29.2

27.6/327.6/3

9.29.2

82.8/982.8/9

9.29.2

Mexican peso conversion is a linear function with respect to US dollar.

CONSTANT RATE OF CHANGE

2.1

dollars

pesos

straight line graph

Mexican peso conversion is a linear function with respect to US dollar.

In summer of 2001, the exchange rate for the Mexican peso was 9.2.

2.1

p(d) = 0.92*d

linear formula: f(x) = b + mx

starting value/y-intercept (b) is 0.

rate of change/slope (m) is 0.92.

Mexican peso conversion is a linear function with respect to US dollar.

In summer of 2001, the exchange rate for the Mexican peso was 9.2.

2.1

Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year .

xx(age)(age)

00 11 22 33 44 55 66 77 88 99 1010

yy(value)(value)

30003000 27002700 24002400 21002100 18001800 15001500 12001200 900900 600600 300300 00

xx 0 to 10 to 1 1 to 21 to 2 0 to 50 to 5 3 to 103 to 10

change in xchange in x 11 11 55 77

yy 3000 to 3000 to 27002700

2700 to 2700 to 24002400

3000 to 3000 to 15001500

2100 to 02100 to 0

change in ychange in y -300-300 -300-300 -1500-1500 -2100-2100

mm -300/1-300/1

-300-300

-300/1-300/1

-300-300

-1500/5-1500/5

-300-300

-2100/7-2100/7

-300-300

CONSTANT RATE OF CHANGE

Value of DJ system is a linear function with respect to age.

2.1

Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year .

straight line graph

Value of DJ system is a linear function with respect to age.

2.1

age (years)

value(dollars)

v(t) = 3000 - 300*t

linear formula: f(x) = b + mx

starting value/y-intercept (b) is 3000 [$].

rate of change/slope (m) is -300 [$ per year].

Jason decides to purchase a $3000 DJ system that has a life expectancy of 10 years. He assumes the value of the equipment will depreciate linearly by the same amount ($300) each year .

Value of DJ system is a linear function with respect to age.

2.1

Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month.

tt(hours)(hours)

00 11 22 1010 2020 100100

billbill(dollars)(dollars)

21.9521.95 21.9521.95 21.9521.95 21.9521.95 21.9521.95 21.9521.95

xx 0 to 10 to 1 1 to 21 to 2 2 to 102 to 10 1 to 201 to 20

change in xchange in x 11 11 88 1919

yy 21.95 to 21.95 to 21.9521.95

21.95 to 21.95 to 21.9521.95

21.95 to 21.95 to 21.9521.95

21.95 to 21.95 to 21.9521.95

change in ychange in y 00 00 00 00

mm 0/10/1

00

0/10/1

00

0/80/8

00

0/190/19

00

CONSTANT RATE OF CHANGE

Monthly bill is a linear function with respect to number of hours used.

2.1

Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month.

STRAIGHT LINE GRAPH

Monthly bill is a linear function with respect to number of hours used.

2.1

time (hours)

bill (dollars)

U(t) = 21.95

linear formula: f(x) = b + mx

starting value/y-intercept (b) is 21.95 [$].

rate of change/slope (m) is 0 [$ per hour].

Monthly bill is a linear function of number of hours spent online.

Under America Online’s Unlimited Usage plan, a member is charged $21.95 per month regardless of the number of hours spent online. Express the monthly bill as a function of the number of hours used in one month.

2.1

Not all straight line graphs are linear functions.

Consider the equation x = 3.

xx 33 33 33 33 33

yy -4-4 -1-1 00 33 55

xx 3 to 33 to 3 3 to 33 to 3 3 to 33 to 3 3 to 33 to 3

change in xchange in x 00 00 00 00

yy -4 to 1-4 to 1 -4 to 0-4 to 0 -1 to 0-1 to 0 0 to 50 to 5

change in ychange in y 55 44 11 55

mm 5/05/0

uu

4/04/0

uu

1/01/0

uu

5/05/0

uu

linear formula: f(x) = b + mx

2.1

An exponential function models any process in which function values change by a fixed ratio or percentage.

The graph of an exponential function is curvy.

An exponential function has the form:

y = f(x) = c * ax

where

f is the name of the function.

c is the starting value or y intercept (f(0)).

a is the growth factor.

2.2

Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million.

tt(20 minute intervals)(20 minute intervals)

00 11 22 33 44 55

PP(number of bacteria)(number of bacteria)

11 22 44 88 1616 3232

tt 0 to 10 to 1 1 to 21 to 2

change in tchange in t 11 11

PP 1 to 21 to 2 2 to 42 to 4

change in Pchange in P 11 22

mm 1/11/1

11

2/12/1

22

NO CONSTANT RATE OF CHANGE [increasing].

2.2

Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million.

tt(20 minute intervals)(20 minute intervals)

00 11 22 33 44 55

PP(number of bacteria)(number of bacteria)

11 22 44 88 1616 3232

Growth factor is 2 [doubling].

ratio of consecutive output valuesratio of consecutive output values

tt P(t+1)/P(t)P(t+1)/P(t)

00 P(1)/P(0) = 2 / 1 = 2P(1)/P(0) = 2 / 1 = 2

11 P(2)/P(1) = 4 / 2 = 2P(2)/P(1) = 4 / 2 = 2

22 P(3)/P(2) = 8 / 4 = 2P(3)/P(2) = 8 / 4 = 2

GRAPH IS CONCAVE UP [increasing rate of change].

Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million.

time (20-minute intervals)

bacteriapopulation

P(t) = 2t

exponential formula: f(x) = c*ax

starting value/y-intercept (c) is 1 [bacteria].growth factor (a) is 2.

Bacteria population is an exponential function of time.

After 8 hours (24 20-minute time intervals):

P(24) = 224 = 16,777,216 bacteria

Harmful kitchen bacteria can double their numbers every 20 minutes. A single bacterium on a wet countertop might in just eight hours, reproduce to nearly 17 million.

tt(years since 1988)(years since 1988)

00 11 22 33

PP(polio cases)(polio cases)

38,00038,000 38000-.25*3800038000-.25*38000= 28500= 28500

28500-.25*2850028500-.25*28500= 21375= 21375

21375-.25*2137521375-.25*21375= 16031= 16031

tt 0 to 10 to 1 1 to 21 to 2

change in tchange in t 11 11

PP 38000 38000 to to

2850028500

28500 28500 to to

2137521375

change in Pchange in P -9500-9500 -7125-7125

mm -9500-9500 -7125-7125

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually.

NO CONSTANT RATE OF CHANGE [increasing].

tt(years since (years since

1988)1988)

00 11 22 33

PP(polio cases)(polio cases)

38,00038,000 38000-.25*3800038000-.25*38000= 28500= 28500

28500-.25*2850028500-.25*28500= 21375= 21375

21375-.25*2137521375-.25*21375= 16031= 16031

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually.

ratio of consecutive output valuesratio of consecutive output values

tt P(t+1)/P(t)P(t+1)/P(t)

00 P(1)/P(0) = 28500 / 38000 = .75P(1)/P(0) = 28500 / 38000 = .75

11 P(2)/P(1) = 21375 / 28500 = .75P(2)/P(1) = 21375 / 28500 = .75

22 P(3)/P(2) = 16031 / 21375 = .7499P(3)/P(2) = 16031 / 21375 = .7499

“growth” factor is 0.75 [decreasing by 25% means 75% remains]

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually.

GRAPH IS CONCAVE UP [increasing rate of change].years since 1988

number of polio cases

During the late twentieth century, WHO adopted as one of its goals the elimination of polio throughout the world. From 1988 to 1996, cases of polio decreased by roughly 25% annually.

P(t) = 38000*(.75)t

exponential formula: f(x) = c*ax

starting value/y-intercept (c) is 38000 [polio cases].growth factor (a) is 0.75.

Number of polio cases is an exponential function of time.

Chapter 2 – Linear and Exponential Functions

HWp81: 1-6, 13-18, 21-23

TURN IN: #13, #16, #22,