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(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
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
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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:**
\[ y=30,000\left(\frac{3}{4}\right)^x \]
\[ y = a \cdot b^x \]
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:**
\[ y=30,000\left(\frac{3}{4}\right)^x \]
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