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.

x(dollar)

0 1 2 3 10

y(peso)

0 9.2 18.4 27.6 92

x 0 to 1 1 to 2 0 to 3 1 to 10

change in x 1 1 3 9

y 0 to 9.2 9.2 to 18.4 0 to 27.6 9.2 to 92

change in y 9.2 9.2 27.6 82.8

m 9.2/19.2

9.2/19.2

27.6/39.2

82.8/99.2

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) = 9.2*d

linear formula: f(x) = b + mx

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

rate of change/slope (m) is 9.2.

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 .

x(age)

0 1 2 3 4 5 6 7 8 9 10

y(value)

3000 2700 2400 2100 1800 1500 1200 900 600 300 0

x 0 to 1 1 to 2 0 to 5 3 to 10

change in x 1 1 5 7

y 3000 to 2700

2700 to 2400

3000 to 1500

2100 to 0

change in y -300 -300 -1500 -2100

m -300/1-300

-300/1-300

-1500/5-300

-2100/7-300

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.

t(hours)

0 1 2 10 20 100

bill(dollars)

21.95 21.95 21.95 21.95 21.95 21.95

x 0 to 1 1 to 2 2 to 10 1 to 20

change in x 1 1 8 19

y 21.95 to 21.95

21.95 to 21.95

21.95 to 21.95

21.95 to 21.95

change in y 0 0 0 0

m 0/10

0/10

0/80

0/190

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.

x 3 3 3 3 3

y -4 -1 0 3 5

x 3 to 3 3 to 3 3 to 3 3 to 3

change in x 0 0 0 0

y -4 to 1 -4 to 0 -1 to 0 0 to 5

change in y 5 4 1 5

m 5/0u

4/0u

1/0u

5/0u

linear formula: f(x) = b + mx

2.1

21

7

7/81

10/81

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.

2.2Harmful 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.

t(20 minute intervals)

0 1 2 3 4 5

P(number of bacteria)

1 2 4 8 16 32

t 0 to 1 1 to 2

change in t 1 1

P 1 to 2 2 to 4

change in P 1 2

m 1/11

2/12

NO CONSTANT RATE OF CHANGE [increasing].

2.2Harmful 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.

t(20 minute intervals)

0 1 2 3 4 5

P(number of bacteria)

1 2 4 8 16 32

Growth factor is 2 [doubling].

ratio of consecutive output values

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

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

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

2 P(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.

t(years since 1988)

0 1 2 3

P(polio cases)

38,000 38000-.25*38000= 28500

28500-.25*28500= 21375

21375-.25*21375= 16031

t 0 to 1 1 to 2

change in t 1 1

P 38000 to 28500

28500 to

21375

change in P -9500 -7125

m -9500 -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].

t(years since

1988)

0 1 2 3

P(polio cases)

38,000 38000-.25*38000= 28500

28500-.25*28500= 21375

21375-.25*21375= 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 values

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

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

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

2 P(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: #1-23

TURN IN: #5, #9, #20 (reference #19), #22,