Question

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

years after purchase	value ($)
1	18,000
2	14,000
3	10,500
4	8,000
5	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 $f(x) = 554x^2 - 5,439x + 24,600$.
• 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: Analyze the first function

For the function \(f(x) = 24512(0.755)^{x}\), when \(x = 0\), \(f(0)=24512(0.755)^{0}=24512\times1 = 24512\approx25000\). When \(x = 1\), \(f(1)=24512\times0.755 = 24512\times\frac{755}{1000}=24512\times0.755 = 18506.56\approx18000\). When \(x = 2\), \(f(2)=24512\times(0.755)^{2}=24512\times0.570025\approx14000\). This function models exponential decay and fits the data well.

Step2: Analyze the second function

For the quadratic function \(y = 554x^{2}-5439x + 24600\), when \(x = 0\), \(y=24600\approx25000\). When \(x = 1\), \(y=554 - 5439+24600=19715
eq18000\). So this quadratic function does not fit the data.

Step3: Analyze the behavior of the exponential function

The exponential function \(y = a\cdot b^{x}\) (\(0 < b<1\)) \(y = 24512(0.755)^{x}\) decreases as \(x\) increases. As \(x\to+\infty\), \(y = 24512(0.755)^{x}\to0\), so it decreases indefinitely.

Step4: Analyze interpolation

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

Step5: Analyze extrapolation

Extrapolation to \(x = 40\) years. A car's value cannot be accurately modeled by the same exponential - decay formula for 40 years. There are practical limitations (e.g., the car will be scrapped, and the formula \(y = 24512(0.755)^{x}\) when \(x = 40\) gives an extremely small non - zero value, but in reality, the value will not follow the formula for such a long time.

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.