Question

Write an exponential decay function in the form f(x)=ab^x for each of Car A and Car B. Explain how you determined the value of b for each function.

Answer 1

Answer:

  A: 8750(0.88^x)

  B: 9995(0.82^x)

Step-by-step explanation:

You want an exponential decay function for cars A and B given their cost and their "decay factor."

Exponential function

In general, an exponential function has the form ...

  value = (initial value)·(growth factor)^x

The growth factor is usually defined as ...

  growth factor = 1 + growth rate

where the "growth rate" is often expressed as a percentage or a fraction.

Your form for f(x) has a=(initial value) and b=(growth factor).

The value of x will be zero at the point where the initial value applies. It will increase by 1 unit for each interval in which the growth factor applies.

Application

Car A

The initial value is presumed to be the Cost. What is called the "growth rate" above is the opposite of what is called the "Decay Factor" in this problem. That is ...

• (initial value) = Cost = 8750
• (growth factor) = 1 - Decay Factor = 1 -0.12 = 0.88
• x = years after 2015

The exponential function is then ...

  f(x) = 8750·(0.88^x)

Car B

For this car, we have ...

• (initial value) = Cost = 9995
• (growth factor) = 1 - Decay Factor = 1 -0.18 = 0.82
• x = years after 2017

The exponential function is then ...

  f(x) = 9995·(0.82^x)