Question

1. you purchase a car from a dealership for $15,000. the trade - in value of the car x years after the purchase is given by the function: (f(x)=15,000 - 300x^{2}) a. what type of function is used to model the data? b. after how many years will the trade - in value be $0? c. in this situation, what is the domain representing? d. in this situation, what is the range representing?

Explanation:

Step1: Identify function type

The function $f(x)=15000 - 300x^{2}$ is a quadratic function since it is in the form $y = ax^{2}+bx + c$ (here $a=- 300$, $b = 0$, $c = 15000$).

Step2: Solve for $x$ when $f(x)=0$

Set $f(x)=0$, so $15000-300x^{2}=0$. First, rewrite it as $300x^{2}=15000$. Then $x^{2}=\frac{15000}{300}=50$. Taking the square - root of both sides, $x=\pm\sqrt{50}=\pm5\sqrt{2}$. Since $x$ represents the number of years after purchase, we take the positive value $x = 5\sqrt{2}\approx7.07$ years.

Step3: Determine the domain

The domain represents the set of all possible input values. Here, $x$ represents the number of years after the car is purchased. Since $x$ cannot be negative (you can't have negative years after purchase), and the car's trade - in value will reach 0 at some point, the domain is $0\leq x\leq5\sqrt{2}$, or in interval notation $[0,5\sqrt{2}]$. In the context of the problem, $x$ represents the number of years since the car was purchased.

Step4: Determine the range

The function $f(x)=15000 - 300x^{2}$ is a downward - opening parabola. When $x = 0$, $f(0)=15000$. When $x = 5\sqrt{2}$, $f(5\sqrt{2})=0$. So the range is $0\leq f(x)\leq15000$, or in interval notation $[0,15000]$. In the context of the problem, the range represents the possible trade - in values of the car.

Answer:

a. Quadratic function
b. $5\sqrt{2}\approx7.07$ years
c. The number of years since the car was purchased, with $0\leq x\leq5\sqrt{2}$
d. The possible trade - in values of the car, with $0\leq f(x)\leq15000$