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

1 answer

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:

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