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AA3-2 Investigation -- modeling exponential growth and decay cloned 1/19/2021

Last updated about 4 years ago
14 questions
In AA3-1 investigation we saw that exponential functions with (h,k)=(0,0) are written in the form:
-- a is the starting value (f(0))
-- b is the rate of growth or decay (what we are multiplying by each time)
-- x is the number of times something occurs. It is often, but not always, an amount of time.

In this investigation, we are going to focus on how to write and solve exponential equations given a function story. To do this we need to do 4 steps:

1) Identify the start value.
2) Determine the rate of growth or decay. Remember 0<b<1 is a rate of decay, and b>1 is a rate of growth.
3) Write the exponential equation.
4) Evaluate f(x) for a given value of x.
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Function story: A person invests $100 and it is growing at a rate of 3% per month.

STEP 1) What is the start value "a"?

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Function story: A person invests $100 and it is growing at a rate of 3% per month.

STEP 2) Determine the rate of growth or decay, "b"?

When the rate of growth or decay is given as a percent increase or decrease (which is often is),
you must write "b" as it relates to change from 100%, then convert it to a decimal.
  • If it is a percent increase, you will still have 100% of what you started with PLUS the percent increase.
100% + 3% = 103% rate of growth
  • If it is a percent decrease, you will have 100% of what you started with MINUS the percent decrease.
100% - 3% = 97% rate of decay

Is it a percent increase or decrease?

1

Now, write the percent as a decimal. To do this divide the rate of growth or decay by 100.

What is the percent increase (103%) written as a decimal. This is "b".

1

Function story: A person invests $100 and it is growing at a rate of 3% per month.

STEP 3) Write the exponential equation in the form:
(Use the equation editor or the ^ will make it an exponent.

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STEP 4) Use your equation to evaluate f(x) at different values of x.

How much will the investment be worth after 4 months? In other words, evaluate f(4).

f(4) = 100(1.03)4

Enter f(4) into the calculator below and it will give you the solution.

1

How much will the investment be worth after 4 months?

It is money so use $ and round to the nearest penny.

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Calculate the value of the investment in 4 years?

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What value of x did you use above? Why?

1

I have 1400 mosquitoes in my backyard. Each day my bug zapper kills 14% of the mosquitoes.

Write the equation for my mosquito population:

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Calculate the mosquitoes population after a week?

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How many mosquitoes will be in my backyard after one week?

Be sure to round to the nearest mosquito (I can't have part of a mosquito) and include "mosquitoes" in your answer. You must answer a story problem with units.
1

Boots bought a house for $120,000, each year its value goes up by 4%. How much is the house worth after 5 years?

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How much is the house worth?

1

Write and solve an exponential story problem.