Question

the value of a car decreases as shown in the table below.
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) = 554x^2 - 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.

Explanation:

Step1: Check the first function

For \(x = 0\), \(f(0)=24512(0.755)^{0}=24512\approx25000\).
For \(x = 1\), \(f(1)=24512\times0.755 = 24512\times\frac{755}{1000}=24512\times0.755 = 18506.56\approx18000\).
For \(x = 2\), \(f(2)=24512\times(0.755)^{2}=24512\times0.570025 = 14073.25\approx14000\).
For \(x = 3\), \(f(3)=24512\times(0.755)^{3}=24512\times0.430368 = 10534.16\approx10500\).
For \(x = 4\), \(f(4)=24512\times(0.755)^{4}=24512\times0.325938 = 8000.03\approx8000\).
For \(x = 5\), \(f(5)=24512\times(0.755)^{5}=24512\times0.246183 = 6034.3\approx6000\). So the function \(y = 24512(0.755)^{x}\) is a good fit.

Step2: Check the second - degree polynomial \(y=554x^{2}-5439x + 24600\)

For \(x = 0\), \(y=24600
eq25000\). So this polynomial is not a good fit.

Step3: Analyze the behavior of the exponential function \(y = 24512(0.755)^{x}\)

Since \(0<0.755 < 1\), as \(x\to+\infty\), \(y = 24512(0.755)^{x}\to0\), so the function decreases indefinitely.

Step4: Analyze interpolation

Interpolation is the process of estimating a value within the range of the known data points. Since \(4.5\) is between \(4\) and \(5\) (the known \(x\) - values), it is reasonable to interpolate the value of the car at \(4.5\) years.

Step5: Analyze extrapolation

Extrapolation to \(x = 40\) years is not reasonable. A car's value will not follow the same pattern for an extremely long - time (in a real - world scenario, the car will be scrapped or its value will approach a minimum non - zero value due to salvage value etc.)

Answer:

• The function that best represents the data is \(f(x)=24,512(0.755)^{x}\).
• The function decreases indefinitely.
• It is reasonable to interpolate to the value of the car at \(4.5\) years.