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=30,000(34)x
y equals 30,000 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=22,500(34)x
y equals 22,500 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline
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Answer 1

To determine the correct exponential equation that fits the given points (0, 30000) and (1, 22500), we can use the general form of an exponential decay equation:

\[ y = a \cdot b^x \]

where:
- \( a \) is the initial value (the value of the car at year 0),
- \( b \) is the decay factor,
- \( x \) is the number of years since purchase.

Given the first point (0, 30000), we have:

\[ a = 30000 \]

Now, we also know from the second point (1, 22500) that:

\[ y = 22500 \]
\[ x = 1 \]

Substituting these values into the equation gives us:

\[ 22500 = 30000 \cdot b^1 \]

To solve for \( b \):

\[ b = \frac{22500}{30000} \]
\[ b = \frac{3}{4} \]
\[ b = 0.75 \]

So the exponential equation representing the car's value over time is:

\[ y = 30000 \left( \frac{3}{4} \right)^x \]

The correct response that matches this equation is:

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

So the correct option is:

**y equals 30,000 left parenthesis Start Fraction 3 over 4 End Fraction right parenthesis superscript x baseline**.