Question

Use the image to answer the 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=0.75(30,000)x
y equals 0.75 left parenthesis 30,000 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=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(34)x
y equals 30,000 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline
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Answer 1

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:

1. **y = 0.75(30,000)^x** (Not correct)
2. **y = 30,000(4/3)^x** (Not correct)
3. **y = 22,500(3/4)^x** (Not correct)
4. **y = 30,000(3/4)^x** (This is correct)

Thus, the correct exponential equation for the graph is:

**y = 30,000(3/4)^x**