To find the exponential equation that passes through the points \((-2, 4)\) and \((3, 972)\), we can assume the general form of the exponential equation is:
\[ y = ab^x \]
where \(a\) is the initial value (the value of \(y\) when \(x = 0\)) and \(b\) is the growth factor.
We can create two equations using the given points:
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For the point \((-2, 4)\): \[ 4 = ab^{-2} \]
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For the point \((3, 972)\): \[ 972 = ab^{3} \]
Now we have a system of two equations:
- \(4 = \frac{a}{b^2}\)
- \(972 = ab^3\)
From the first equation, we can express \(a\) in terms of \(b\): \[ a = 4b^2 \]
Now, substitute this expression for \(a\) into the second equation: \[ 972 = (4b^2)b^3 \] \[ 972 = 4b^{5} \]
Now, solve for \(b\): \[ b^5 = \frac{972}{4} = 243 \]
To solve for \(b\), take the fifth root of \(243\): \[ b = 243^{1/5} = 3 \]
Now that we have \(b\), we can find \(a\) using \(a = 4b^2\): \[ a = 4(3^2) = 4 \times 9 = 36 \]
Now we have both \(a\) and \(b\): \[ a = 36, \quad b = 3 \]
So, the exponential equation that contains the points \((-2, 4)\) and \((3, 972)\) is: \[ y = 36 \cdot 3^x \]
The final equation is: \[ \boxed{y = 36 \cdot 3^x} \]