Exponential Equations

Exponential Equations

 $y=a\left(b\right)^2$  

 $y=a\left(b\right)^x$  

In this exponential equation the a is the starting value, just like a is the first letter in the alphabet. The b is called the growth factor or decay factor depending on if it is going up or down, (1 + r) if growth, (1 - r) if decay.  The r is the percentage rate.  The x it the number of cycles it goes through.

Let's say we are looking at the value of a car. As a car gets older its value depreciates, goes down.

• The a value is the original value of the car.

• The b value is (1 - r) where r is the percentage rate it decreases by.

• The x is how many years have passed since you purchased the car.

Purchase price is $19,000

• It depreciates at 15% per year

• So b = (1 - .15) = .85

• That means that the car is worth 85% of its original value

• x will be 8 since they want the value of the car after 8 years

Multiple Choice

Samantha's hair was known to grow very rapidly. It began at a length of 6 in and grew at a rate of 14% a week. What is the a value, starting value?

• 6

• 14%

• (1 + .14)

• (1 - .14)

Multiple Choice

Samantha's hair was known to grow very rapidly. It began at a length of 6 in and grew at a rate of 14% a week. What is the b value, growth factor?

• 6

• 14%

• (1 + .14)

• (1 - .14)

Multiple Choice

Write an equation that models the following situation:

Samantha's hair was known to grow very rapidly. It began at a length of 6 in and grew at a rate of 14% a week.

• y=6(0.14)^x

• y=6(1+14)^x

• y=6(1.14)^x

• y=6(0.86)^x

Growth or Decay

Exponential growth is looks like a curve increasing (gets higher) left to right. Exponential decay looks like a curve decreasing (gets smaller) left to right.

Multiple Choice

Is the pictured graph growth or decay?

• Growth

• Decay

• I don't know yet.

Multiple Choice

Is the pictured graph growth or decay?

• Growth

• Decay

• I don't know yet.

How can I tell if it is growth or decay in an equation?

When b > 1 it is growth (1 + r). When b < 1 then it is decay (1 - r).

Multiple Choice

Denise has $80 in a savings account that earns 10% interest, compounded annually.

What is the starting value (a)?

• 10%

• $80

• ( 1 + 10%) = 1.1

• (1 - 10%) = .9

Multiple Choice

Denise has $80 in a savings account that earns 10% interest, compounded annually.

What is her growth factor (b)?

• 10%

• $80

• (1 + 10%) = (1.1)

• (1 - 10%) = (.9)

Multiple Choice

Denise has $80 in a savings account that earns 10% interest, compounded annually.

To the nearest cent, how much will she have in 3 years?

• $y\ =\ 80\left(10\%\right)^3$

• $y=80\left(1.1\right)^3$

• $y=80\left(.9\right)^3$

• I don't know yet.

Multiple Choice

Which of the following functions shows an initial amount of $15 and an increase of 35% each year?

• y = 15(35)^x

• y = 15(0.35)^x

• y = 15(1.35)^x

• y = 35(1.15)^x

Multiple Choice

What do you think?

You bought a Boston Whaler in 2014 for $12,500. The boat's value depreciates by 7% a year. How much is the boat worth in 2021?

• $11625

• $7521.26

• $20072.27

• $875

Multiple Choice

An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s system decreases by about 29%.

What is the b value for this problem?

• 1.29

• .71

• .29

• 1.71

Multiple Choice

An adult takes 400 mg of ibuprofen. Each hour, the amount of ibuprofen in the person’s system decreases by about 29%.

What is the equation we should use to solve it?

• $y=400\left(.29\right)^x$

• $y=400\left(1.29\right)^x$

• $y=400\left(.71\right)^x$
  

• I don't know yet

Multiple Choice

The population of Winnemucca, Nevada, can be modeled by P=6191(1.04)^t where t is the number of years since 1990. What percent (r) did the population increase each year?

• 4%

• 6191

• I don't know yet

• 104%

Multiple Choice

The population of Winnemucca, Nevada, can be modeled by P=6191(1.04)^t where t is the number of years since 1990. What was the population in 1990?

• 4%

• 104%

• I don't know yet

• 6191