Use the image to answer the question.

Question

A descending curve with an arrow on the right end passes through 10 points plotted on a coordinate plane. The x-axis is labeled years since purchase, and the y-axis is labeled value of car.

Given the graph of the exponential equation representing the value of a car since purchase, which option is the correct exponential equation for the graph if the graph goes through the points (0, 30000) and (1, 22500) ?

(1 point)
Responses

y=22,500(34)x
y equals 22,500 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline

y=30,000(43)x
y equals 30,000 left parenthesis Start Fraction 4 over 3 End Fraction right parenthesis superscript x baseline

y=0.75(30,000)x
y equals 0.75 left parenthesis 30,000 right parenthesis superscript x baseline

y=30,000(34)x
y equals 30,000 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline

Answer 1

To find the correct exponential equation for the value of a car since its purchase, we can use the information provided from the points (0, 30000) and (1, 22500).

1. Determine the initial value (at x=0): At \( x = 0 \), the value of the car is \( y = 30000 \). Therefore, the equation will have the form: \[ y = 30000 \cdot a^x \]

2. Use the second point (1, 22500): At \( x = 1 \), the value of the car is \( y = 22500 \). Plugging this into the equation gives us: \[ 22500 = 30000 \cdot a^1 \]

3. Solve for \( a \): \[ a = \frac{22500}{30000} = \frac{3}{4} \]

4. Formulate the exponential equation: Now substituting \( a \) back into the original equation, we have: \[ y = 30000 \left(\frac{3}{4}\right)^x \]

This matches the fourth option given in the responses: \[ y = 30,000 \left(\frac{3}{4}\right)^x \]

Thus, the correct exponential equation representing the graph is: \[ \boxed{y = 30,000 \left(\frac{3}{4}\right)^x} \]