8.1Exponential Growth

Learning Targets

Students should be able to…

Graph exponential growth functions.

Go over Chapter 7 test

Warm-up/Introduction

http://math.rice.edu/~lanius/pro/rich1.html

Exponential function-

involves the expression bx where b is a positive number other than 1.

Base

is the number b in an exponential function.

Asymptote-

is a line that a graph approaches as you move away from the origin. y = ab

Exponential growth function-

is an exponential function in the form where a > 0 and b > 1.

Graph exponential growth functions using y = abx - h + k

x y

0 3

1 6

Graph y = 3·2x

Make a table with x values 0 and 1

Plot the points and draw a curve that runs close to the x-axis and passes through the two points

D: all real

R: y > 0

Asymptote y = 0

Graph y = 3·4x -1

x y

0 3

1 12

x y

1 3

2 12

Make a table with x values 0 and 1

Begin by looking at the un shifted graph of y= 3·4x

Shift the two points right 1

D: all real R: y > 0

Asymptote y = 0

Graph y = 4·2 x - 3 + 1

x y

0 4

1 8

x y

3 5

4 9

Begin by looking at the un shifted graph of y= 4·2x

Make a table with x values 0 and 1

Shift the two points right 3 and up 1

D: all real

R: y > 1

Asymptote y = 1

2. Use exponential growth models in real life

When a real-life quantity increases by a fixed percent each year (or other time period), the amount y of the quantity after t years can be modeled by this equation:

y = a(1 + r)t

In this model, a is the initial amount and r is the percent increase expressed as a decimal. The quantity 1 + r is called the growth factor.

In 1980 wind turbines in Europe generated about 5 gigawatt-hours of energy. Over the next 15 years, the amount of energy increased by about 59% per year.

1. Write a model giving the amount E (gigawatt-hours) of energy t years after 1980. About how much wind energy was generated in 1984?

E=5(1.59)t

About 32 hours

2. Graph the model.

3. Estimate the year when 80 gigawatt-hours of energy were generated? About 5.98 near the end of 1985

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

V = 54(1.05)t

Compound Interest

Consider an initial principal P deposited in an account that pays interest at an annual rate r (expressed as a decimal) compounded n times per year. The amount A in the account after t years can be modeled by

You deposit $1500 in an account that pays 6% annual interest.

Find the balance after 1 year if the interest is compounded A. Annually

B. Semi-annually

C. Quarterly

nt

n

rPA )+1(=

1•1

)1

06.+1(1500=A =1590

1•2

)2

06.+1(1500=A =1591.35

=1592.051•4

)4

06.+1(1500=A

1. 2.

y = 2x