To find the correct exponential equation representing the value of a car since purchase, we can start with the general form of an exponential decay function, which is:
\[ y = a \cdot b^x \]
Where:
- \( y \) is the value of the car at \( x \) years since purchase,
- \( a \) is the initial value of the car (at \( x = 0 \)),
- \( b \) is the decay factor,
- \( x \) is the number of years since purchase.
From the problem, we have:
- At \( x = 0 \), the value \( y = 30,000 \). This means \( a = 30,000 \).
- At \( x = 1 \), the value \( y = 22,500 \).
Now we can write the equation based on the information we have:
\[ y = 30,000 \cdot b^x \]
Substituting the second point (1, 22,500) into the equation:
\[ 22,500 = 30,000 \cdot b^1 \]
To find \( b \), we divide both sides by 30,000:
\[ b = \frac{22,500}{30,000} = \frac{3}{4} = 0.75 \]
So, we can substitute back into the equation:
\[ y = 30,000 \cdot \left(\frac{3}{4}\right)^x \]
Now we check the options provided:
- y = 0.75(30,000)^x (Not correct)
- y = 30,000(4/3)^x (Not correct)
- y = 22,500(3/4)^x (Not correct)
- y = 30,000(3/4)^x (This is correct)
Thus, the correct exponential equation for the graph is:
y = 30,000(3/4)^x