Question

Write an exponential equation that contains the points negative 2, 4 and 3, 972 equation is y equals

Answer 1

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:

1. **Substituting the points into the equation:**

For the point \((-2, 4)\):

\[
4 = ab^{-2} \quad \text{(1)}
\]

For the point \((3, 972)\):

\[
972 = ab^{3} \quad \text{(2)}
\]

2. **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:

\[
b^{5} = \frac{972}{4} = 243
\]

3. **Solving for \( b \):**

Taking the fifth root:

\[
b = 243^{1/5}
\]

Since \( 243 = 3^5 \), we have:

\[
b = 3
\]

4. **Finding \( a \):**

Now substitute \( b = 3 \) back into equation (3):

\[
a = 4(3^{2}) = 4 \cdot 9 = 36
\]

5. **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:

\[
\boxed{y = 36 \cdot 3^x}
\]