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

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

Answer 1

To determine the correct exponential equation for the graph that passes through the points (0, 30000) and (1, 22500), we can start by using the general form of an exponential decay function:

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

where:

• \( a \) represents the initial value (the value when \( x = 0 \)),
• \( b \) is the decay factor, and
• \( x \) is the number of years since purchase.

From the point (0, 30000):

• When \( x = 0 \), \( y = 30000 \), hence \( a = 30000 \).

From the point (1, 22500):

• When \( x = 1 \), \( y = 22500 \).
• Substituting into the equation:

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

To solve for \( b \):

1. Divide both sides by 30000:

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

Thus, the equation can be rewritten as:

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

Now, we can look at the provided options:

1. \( y = 30,000(4/3)^x \)
2. \( y = 30,000(3/4)^x \) (this matches our derived equation)
3. \( y = 0.75(30,000)^x \)
4. \( y = 22,500(3/4)^x \)

The correct exponential equation for the graph is:

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