Question

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

Explanation:

Step1: Recall Exponential Decay Formula

The general formula for exponential decay is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( x \) is time. Here, \( a = 12000 \) and \( r = 0.12 \), so \( 1 - r = 1 - 0.12 = 0.88 \). Thus, the function should be \( f(x)=12000(0.88)^x \).

Step2: Analyze Asymptotes of Exponential Functions

Exponential functions of the form \( f(x)=ab^x \) (where \( b>0, b
eq1 \)) have a horizontal asymptote at \( y = 0 \) (since as \( x\to\pm\infty \), the function approaches 0, depending on \( b \)). Vertical asymptotes are not typical for exponential functions (they are for rational functions, etc.). So the first option has the correct function and horizontal asymptote.

Answer:

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