Objectives:1. Understand the exponential

growth/decay function family. 2. Graph exponential growth/decay

function using y = abx – h + k3. Use exponential function to models

in real life.

Vocabulary:Euler’s number, natural base

5.3 Exponential Growth/Decay

Question 1: Interest of a bank AccountYou deposit P dollars in the bank and receive an interest rate of r compounded annually for t year.

Initial Asset A0 = P

The end of 1st year: A1= P + P·r = P(1 + r)

The end of 2nd year: A2= A1 + A1·r = A1(1 + r) = P(1 + r)(1 + r) = P(1 + r)2

The end of 3rd year: A3= A2 + A2·r = A2(1 + r) = P(1 + r)2 (1 + r) = P(1 + r)3

:::

t-th year: At = P(1 + r)t

Since r > 0, then 1 + r > 1. Denote b = 1 + r, then b > 1. So the above model can be written as:

At = P ·bt (b > 1)

Question 2: Cell splitYou have C cells and each cell will split to 2 new cells. After n splits how many cells you will have?

Initial Cell: C0 = C = C ·20

1st split: C1= C ·2 = C ·21

2nd split: C2= C1·2 = C ·22

3rd split: C3= C2·2 = C ·23

::

n-th split: Cn= C ·2n

Cn = C ·bn (b = 2 > 1)

The observation we had from the above 2Questions attracts us to have a deeper studyto some function like:

f (x) = a bx

where a ≠ 0, b > 0 and b ≠ 1

Definition Exponential FunctionThe function is of the form: f (x) = a bx, where a

≠ 0, b > 0 and b ≠ 1, x R.

Definition Exponential Growth FunctionThe function is of the form: f (x) = a bx, where a

> 0, b > 1, x R.

Definition Exponential Decay FunctionThe function is of the form: f (x) = a bx, where a

> 0, 0 < b < 1, x R.

Summary

f (x) = a b x

a ≠ 0, b > 0, b ≠ 1

x R Exponential Function

a > 0, b > 0, b ≠ 1 Exponential Growth ( b > 1)

Exponential Decay ( 0 < b < 1)

f (x) = a b x

a > 0, b > 0, b ≠ 1

x R

Domain x R x R

Range y > 0 y > 0

aa

Challenge Question 1: In the definition, why b > 0 and b ≠ 1?

Function Family -- functions whose graphs are identical if the graphs are shifted some units horizontally and/or vertically.

Activity

Using the graphing calculator to graph:a) y = 2 · 3x

b) y = 2 · 3x+1 - 5

Graph b) is -- shifted 1 unit to the left of a) (horizontally)-- shifted 5 unit down of a) (vertically)-- “new” y-intercept is at (-1, -3)-- “new” asymptotes are line y = - 5 and line x = -1.

New center is at (-1, - 5)

Procedures

1) Find the new center (h, k)2) Draw 2 perpendicular dash lines to represent new x-axis and new y-axis3) Label a + k as “new” y-intercept4) Treat horizontal and vertical dash line as “new” x-axis and “new” y-axis.

c) Example 1 Graph y = 2x

y = (1/2)x

y = -3 · 2x

d) y = -2 (1/3)x

Conclusions:

1) Graph a) is an exponential growth function graph and y – intercept is 1.2) Graph b) is an exponential decay function graph and y – intercept is 1.3) Graphs c) and d) are not an exponential growth or decay function.

Challenge Question 2: The product of two exponential growth(or decay) functions is still an exponential growth(or decay) function? Justify your answer.

Suppose that y1 = a1 b1x and y2 = a2 b2

x are exponential growth function, where a1, a2 R+, b1, b2 R+, b1 > 1, b2 > 1

then since b1 b2 > 1

y1y2 = a1 a2 b1x b2

x = a1 a2 (b1b2 )x

is also an exponential growth function.

Challenge Question 3: Given an exponential growth(or decay) function y = a bx, can you construct an exponential decay(or growth) function?

Given that y1 = a1 b1x is an exponential growth

function, take a2 = a1 R+, b2 = 1/ b1 R+, and then 0 < b2 < 1

Therefore, y2 = a2 b2

x

is an exponential decay function.

Example 2 Graph y = -3 · 2x

Example 3 Graph y = 3 · 4x-1

Example 4 Graph y = 4 · 3x-2 + 1

Example 5 Graph y = 3 · (1/2)x

Example 6 Graph y = -3 · (1/4)x-1

Example 7 Graph y = 4 · (1/2)x-3 + 2

The variations of exponential growth/decay function model is very practical in a wide variety of application problems.

Growth: y = a (1 + r ) t

Decay: y = a (1 – r ) t

Note: the rate r is measured the same time period as t, where a stands for the initial amount.

Example 8 You purchase a baseball card for $54. If it increases each year by 5%, write an exponential growth model.

Example 9 In 1980 wind turbines in Europegenerated about 5 gigawatt-hours of energy. Over the next 15 years, the amount of energy increased by about 5.9% per year.a) Write a model giving the amount E (gigawatt-hours) of energy and t years after 1980. About how much wind energy was generated in 1984?

b) Graph the model.

c) Estimate the year when 8.0 gigawatt-hours of energy were generated?

Example 10 You drink a beverage with 120milligrams of caffeine. Each hour, the amount c of caffeine in your system decreases by about 12%.a) Write an exponential decay model.

b) How much caffeine remains in our system after 3 hours?

c) After how many hours, the amount caffeine in your system is 50 milligrams of caffeine?

Example 11 You have a new computer for $2100.The value of the computer decreases by about 30% annually.a) Write an exponential decay model for the value of the computer. Use the model to estimate the value after 2 years.

b) Graph the model.

c) Estimate when the computer will be worth $500?

Assignment:5.1 P469 #14-40 even5.1 P469 #43-48, 56-66

5.2 P 477 #11-52

5.1/5.2 Exponential Growth/Decay