Write an exponential function \( y=a b^{x} \) whose graph passes through \( (3,56) \) and \( (5,224) . \quad y=a b^{x} \)

To find the exponential function \( y = a b^{x} \) that passes through the points \( (3, 56) \) and \( (5, 224) \), we can start by creating a system of equations using these points. Plugging in the point \( (3, 56) \): \[ 56 = a b^{3} \] (1) Plugging in the point \( (5, 224) \): \[ 224 = a b^{5} \] (2) Next, we can solve equations (1) and (2) simultaneously. Dividing equation (2) by equation (1) to eliminate \( a \): \[ \frac{224}{56} = \frac{a b^{5}}{a b^{3}} \] which simplifies to: \[ 4 = b^{2} \] So, we can take the square root of both sides: \[ b = 2 \quad \text{(since the base must be positive)} \] Now we can substitute \( b = 2 \) back into equation (1) to find \( a \): \[ 56 = a (2^{3}) \] \[ 56 = a \cdot 8 \] Now divide both sides by 8: \[ a = 7 \] So, we have \( a = 7 \) and \( b = 2 \). Therefore, the function is: \[ y = 7 \cdot 2^{x} \] Feel free to verify by plugging in the original points!