Question
Write an exponential equation that contains the points (-2, 4) and (3, 972)
Answers
Answered by
GPT-4o mini
To create an exponential equation that fits the points \((-2, 4)\) and \((3, 972)\), we can use the general form of an exponential function:
\[
y = ab^x
\]
where \(a\) is the initial value (when \(x = 0\)), \(b\) is the base of the exponential function, and \(x\) is the independent variable.
Given the two points, we can plug them into the equation to form a system of equations.
1. For the point \((-2, 4)\):
\[
4 = ab^{-2}
\]
2. For the point \((3, 972)\):
\[
972 = ab^{3}
\]
Now we have the following system of equations:
1. \(ab^{-2} = 4\)
2. \(ab^{3} = 972\)
From the first equation, we can solve for \(a\):
\[
a = 4b^{2}
\]
Now we can substitute \(a\) into the second equation:
\[
972 = (4b^{2})b^{3}
\]
\[
972 = 4b^{5}
\]
\[
b^{5} = \frac{972}{4} = 243
\]
Now we take the fifth root to find \(b\):
\[
b = 243^{1/5} = 3
\]
Next, we substitute \(b\) back into the equation for \(a\):
\[
a = 4(3^{2}) = 4 \cdot 9 = 36
\]
Now we have \(a = 36\) and \(b = 3\). Therefore, the exponential equation is:
\[
y = 36(3^{x})
\]
This equation can be verified by plugging the original points back into the equation to check if they satisfy it:
1. For \((-2, 4)\):
\[
y = 36(3^{-2}) = 36 \cdot \frac{1}{9} = 4
\]
2. For \((3, 972)\):
\[
y = 36(3^{3}) = 36 \cdot 27 = 972
\]
Both points satisfy the equation, confirming that the correct exponential equation is:
\[
y = 36(3^{x})
\]
\[
y = ab^x
\]
where \(a\) is the initial value (when \(x = 0\)), \(b\) is the base of the exponential function, and \(x\) is the independent variable.
Given the two points, we can plug them into the equation to form a system of equations.
1. For the point \((-2, 4)\):
\[
4 = ab^{-2}
\]
2. For the point \((3, 972)\):
\[
972 = ab^{3}
\]
Now we have the following system of equations:
1. \(ab^{-2} = 4\)
2. \(ab^{3} = 972\)
From the first equation, we can solve for \(a\):
\[
a = 4b^{2}
\]
Now we can substitute \(a\) into the second equation:
\[
972 = (4b^{2})b^{3}
\]
\[
972 = 4b^{5}
\]
\[
b^{5} = \frac{972}{4} = 243
\]
Now we take the fifth root to find \(b\):
\[
b = 243^{1/5} = 3
\]
Next, we substitute \(b\) back into the equation for \(a\):
\[
a = 4(3^{2}) = 4 \cdot 9 = 36
\]
Now we have \(a = 36\) and \(b = 3\). Therefore, the exponential equation is:
\[
y = 36(3^{x})
\]
This equation can be verified by plugging the original points back into the equation to check if they satisfy it:
1. For \((-2, 4)\):
\[
y = 36(3^{-2}) = 36 \cdot \frac{1}{9} = 4
\]
2. For \((3, 972)\):
\[
y = 36(3^{3}) = 36 \cdot 27 = 972
\]
Both points satisfy the equation, confirming that the correct exponential equation is:
\[
y = 36(3^{x})
\]
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