To find an exponential equation of the form \( y = ab^x \) that passes through the points \((-2, 4)\) and \((3, 972)\), we can follow these steps:
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Substituting the points into the equation:
For the point \((-2, 4)\):
\[ 4 = ab^{-2} \quad \text{(1)} \]
For the point \((3, 972)\):
\[ 972 = ab^{3} \quad \text{(2)} \]
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Rearranging equations to express \( a \) in terms of \( b \):
From equation (1):
\[ a = 4b^{2} \quad \text{(3)} \]
Now, substitute equation (3) into equation (2):
\[ 972 = (4b^{2})b^{3} \]
This simplifies to:
\[ 972 = 4b^{5} \]
Dividing both sides by 4 gives:
\[ b^{5} = \frac{972}{4} = 243 \]
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Solving for \( b \):
Taking the fifth root:
\[ b = 243^{1/5} \]
Since \( 243 = 3^5 \), we have:
\[ b = 3 \]
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Finding \( a \):
Now substitute \( b = 3 \) back into equation (3):
\[ a = 4(3^{2}) = 4 \cdot 9 = 36 \]
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Writing the final exponential equation:
Now we have both \( a \) and \( b \):
\[ y = ab^x = 36 \cdot 3^x \]
Thus, the exponential equation that contains the points \((-2, 4)\) and \((3, 972)\) is:
\[ \boxed{y = 36 \cdot 3^x} \]