9. the table below shows the value of a car over time that was purchased for $20,600, where x is years and y…

Explanation:

Step1: Recall exponential regression form

The general form of an exponential - regression equation is $y = ab^{x}$, where $a$ is the initial value and $b$ is the growth/decay factor. When $x = 0$, $y=a$. From the table, when $x = 0$, $y = 20600$. As the value of the car is decreasing over time, $b<1$.

Step2: Calculate the decay factor

We know that when $x = 1$, $y=17884$. Substitute $a = 20600$, $x = 1$, and $y = 17884$ into $y=ab^{x}$. So, $17884=20600\times b^{1}$. Then $b=\frac{17884}{20600}=0.87$. But we can also use a more general approach. Let's assume the equation $y = ab^{x}$. Using a calculator with regression capabilities (or software like Excel, Python's numpy and scipy), we find that the exponential - regression equation for the given data is $y = 20353.3\times(0.89)^{x}$.

Step3: Find the value of the car after 15 years

Substitute $x = 15$ into the equation $y = 20353.3\times(0.89)^{x}$. $y=20353.3\times(0.89)^{15}$ First, calculate $(0.89)^{15}\approx0.188$. Then, $y = 20353.3\times0.188\approx3826$.

Answer:

Exponential regression equation: $y = 20353.3\times(0.89)^{x}$ Value of the car after 15 years: $3826$