Question

modeling with exponential decay functions
a new automobile loses 12% of its value every year. the initial value is $12,000. which time association for the graph of the function that represents the value of the car is select a choice?
$f(x)=12,000(0.88)^x$ with a horizontal asymptote of $y=0$
$f(x)=12,000(0.12)^x$ with a horizontal asymptote of $y=0$
$f(x)=(12,000 - 0.88)^x$ with a horizontal asymptote of $y=0$
$f(x)=12,000(1.12)^x$ with a horizontal asymptote of $y=0$

Explanation:

Step1: Identify decay factor

Since the value decreases by 12% yearly, the remaining value percentage is $100\% - 12\% = 88\% = 0.88$. The initial value is $\$12,000$, so the exponential decay function has the form $f(x) = \text{Initial Value} \times (\text{Decay Factor})^x$.

Step2: Confirm asymptote

For exponential decay functions of the form $f(x) = ab^x$ where $0 < b < 1$, the horizontal asymptote is $y=0$, as the value approaches 0 as $x$ increases.

Answer:

$f(x) = 12,000(0.88)^x$ with a horizontal asymptote of $y = 0$