Question

modeling with exponential decay functions

jason estimates that his car loses 12% of its value every year. the initial value is $12,000. which best describes the graph of the function that represents the value of the car after x years?

\\(f(x) = 12,000(0.12)^x\\), with a horizontal asymptote of \\(y = 0\\)
\\(f(x) = 12,000(1.12)^x\\), with a vertical asymptote of \\(x = 0\\)
\\(f(x) = 12,000(0.88)^x\\), with a horizontal asymptote of \\(y = 0\\)
\\(f(x) = (12,000 \cdot 0.88)^x\\), with a vertical asymptote of \\(x = 0\\)

Explanation:

Determine the decay factor

$$ b = 1 - r = 1 - 0.12 = 0.88 $$

Formulate the exponential decay function

$$ f(x) = a \cdot b^x = 12,000(0.88)^x $$

Identify the asymptote of the function

$$ \lim_{x \to \infty} 12,000(0.88)^x = 0 \implies \text{Horizontal asymptote: } y = 0 $$

Answer:

• (A) \(f(x) = 12,000(0.12)^x\), with a horizontal asymptote of \(y = 0\)
• (B) \(f(x) = 12,000(1.12)^x\), with a vertical asymptote of \(x = 0\)
• (C) \(f(x) = 12,000(0.88)^x\), with a horizontal asymptote of \(y = 0\) (Correct answer)
• (D) \(f(x) = (12,000 \cdot 0.88)^x\), with a vertical asymptote of \(x = 0\)