Question

The value of a car decreases as shown in the table below.

A 2-column table with 6 rows. The first column is labeled years after purchase with entries 0, 1, 2, 3, 4, 5. The second column is labeled value (dollar sign) with entries 25,000; 18,000; 14,000; 10,500; 8,000; 6,000.
Which statements are true? Check all that apply.

The function that best represents the data is f(x) = 24,512(0.755)x.
The function that best represents the data is f(x) = 554x2 – 5,439x + 24,600.
The function decreases indefinitely.
It is reasonable to interpolate to the value of the car at 4.5 years.
It is reasonable to extrapolate to 40 years.

Answer 1

True:
- The function that best represents the data is f(x) = 24,512(0.755)^x.
(This exponential fits the six data points very closely — e.g., it predicts about \$6,000 at 5 years and matches the other values within a few hundred dollars.)
- The function decreases indefinitely.
(Mathematically the exponential model keeps decreasing for all x ≥ 0 and approaches 0 as x → ∞.)

- It is reasonable to interpolate to the value of the car at 4.5 years.
(4.5 is inside the data range [0,5], so estimating between the 4- and 5-year values is reasonable.)

False:
- The function f(x) = 554x^2 – 5,439x + 24,600.
(This upward-opening quadratic does not match the later data well and would eventually predict increasing value, which is not appropriate.)
- It is reasonable to extrapolate to 40 years.
(40 years is far outside the data range; extrapolating that far is not reliable for a real car’s value.)