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

Answer 1

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

\[ y = a(b^x) \]

where \( a \) is the initial value and \( b \) is the decay factor.

From the point (0, 30000), we know that when \( x = 0 \):

\[ y = a(b^0) = a(1) = a \] Thus, \( a = 30000 \).

Now, using the point (1, 22500), we can substitute \( x = 1 \) and \( y = 22500 \):

\[ 22500 = 30000(b^1) \]

To find \( b \), we can solve for it:

\[ 22500 = 30000b \]

Dividing both sides by 30000 gives:

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

Now we can write the equation using the values we have found:

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

Thus, the correct exponential equation for the graph is:

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

Therefore, the correct option is:

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