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:

\[
y = 36(3^{x})
\]
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