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(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=22,500(34)x

1 answer

To find the correct exponential equation for the given points (0, 30000) and (1, 22500), we can start by identifying 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 \( x = 0 \)).
  • \( b \) is the decay factor per year.
  • \( x \) is the number of years since purchase.

From the points provided:

  1. At \( x = 0 \), the value of the car is \( y = 30000 \). This gives us \( a = 30000 \). So, our equation becomes: \[ y = 30000 \cdot b^x \]

  2. At \( x = 1 \), the value of the car is \( y = 22500 \). Substituting \(x = 1\) and \(y = 22500\) into the equation: \[ 22500 = 30000 \cdot b^1 \]

To solve for \(b\), we can rearrange the equation: \[ b = \frac{22500}{30000} = \frac{3}{4} \]

Now, we can substitute both \( a \) and \( b \) back into the equation: \[ y = 30000 \cdot \left(\frac{3}{4}\right)^x \]

Now, examining the response options:

  1. \( y=0.75(30,000)x \): This is incorrect because it doesn't follow the exponential form.
  2. \( y=30,000\left(\frac{3}{4}\right)^x \): This matches our derived equation.
  3. \( y=30,000\left(\frac{4}{3}\right)^x \): This is incorrect as it suggests an increase rather than decay.
  4. \( y=22,500\left(\frac{3}{4}\right)^x \): This is incorrect for the same reason—while it uses the decay factor, it incorrectly uses a different initial value.

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