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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