Final answer:
The function representing the value of a car after x years given a yearly depreciation of $1,200 and a value of $22,600 after 8 years is y = -1,200x + 32,200. This accounts for the depreciation slope and the original value of the car when new.
Step-by-step explanation:
To write a function that represents the value of a car after x years, we need to know the initial value of the car and the rate at which it depreciates every year. Given that the car depreciates at a rate of $1,200 per year and is worth $22,600 after 8 years, we can set up the function as follows:
Let y be the value of the car after x years. If x = 0 represents the year when the car was new, then x = 8 corresponds to a car valued at $22,600. Since the car depreciates at the rate of $1,200 per year, the car's value decreases linearly over time. Therefore, the slope (m) of this line is -1,200, as it loses that amount in value each year.
First, we need to find the y-intercept (known as b in the equation y = mx + b). To do so, we can use the information that after 8 years (x = 8), the car is worth $22,600. Applying this to the equation gives us:
$22,600 = -1,200(8) + b
$22,600 = -9,600 + b
b = $22,600 + $9,600
b = $32,200
So, the y-intercept is $32,200. This is the initial value of the car when it was new. Now that we have both m (the slope) and b (the y-intercept), we can write the function for the car value y after x years as:
y = -1,200x + 32,200
Therefore, the function representing the value of the car after x years is y = -1,200x + 32,200.