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Exponential Decay

Rational functions with x as an exponent in the denominator

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Teacher Contributed

Real World Applications – Algebra I


What’s a car worth?

Student Exploration

How much is a car worth? You’ve already done several activities in this concept that calculate a car’s worth. You already know that as time passes, a car’s value depreciates. How can you calculate the depreciation rate if it’s not given to you?

  1. Choose a car that you, a friend, a relative, or a neighbor has. Make sure you can talk to the car’s owner. You have a lot of questions to ask them!
    1. What year did you purchase this car? How much did you purchase this car for? How many years have you had this car?
    2. What is the car’s mileage?
  2. Once you have this information, use the internet to find out how much this car is worth present day.
    1. You can use the Kelly Blue Book website: www.kbb.com
  3. Create an equation using the information you have.
    1. What is the initial value?
    2. What is the amount of time that has passed?
    3. What is the value of the car for that amount of time passed?
  4. What are some methods to find out the depreciation rate?
    1. One of the methods is using a calculator. You can use a the Guess and Check method to find out different values of “\begin{align*}b\end{align*}” that would make the equation true.
    2. How could you find the “\begin{align*}b\end{align*}” value algebraically? Show all steps. You will need a scientific or graphing calculator to do this.
  5. Graph this relationship. What values are on the \begin{align*}x-\end{align*}axis and \begin{align*}y-\end{align*}axis? What values make sense on these axes?
  6. Is there ever a time when the car will have no value? If so, when will this happen? If not, why not?

Extension Investigation

7. Use the internet to find the value of a different car in two different years.
8. Use this information to find the initial value of the car.
a. Need help? Get the unknown variable on one side of the equation for both equations, and set the two equations equal to each other.
b. You will need to use logarithms to solve this!

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