Ryan decides to find the exponential function that passes through the points \( (3,135) \) and \( (1,60) \). Use these points to write the equation of his function of the form \( f(x)=a b^{x} \).

To find the exponential function \( f(x) = a \cdot b^{x} \) that passes through the points \( (3, 135) \) and \( (1, 60) \), follow these steps: 1. **Set Up Equations Using the Given Points:** - For the point \( (1, 60) \): \[ f(1) = a \cdot b^{1} = 60 \implies a \cdot b = 60 \quad \text{(Equation 1)} \] - For the point \( (3, 135) \): \[ f(3) = a \cdot b^{3} = 135 \quad \text{(Equation 2)} \] 2. **Divide Equation 2 by Equation 1 to Eliminate \( a \):** \[ \frac{a \cdot b^{3}}{a \cdot b} = \frac{135}{60} \implies b^{2} = \frac{9}{4} \] Solving for \( b \): \[ b = \sqrt{\frac{9}{4}} = \frac{3}{2} \] 3. **Find \( a \) Using Equation 1:** \[ a \cdot \frac{3}{2} = 60 \implies a = 60 \cdot \frac{2}{3} = 40 \] 4. **Write the Final Exponential Function:** \[ f(x) = 40 \cdot \left( \frac{3}{2} \right)^{x} \] **Final Answer:** \[ f(x) = 40 \times \left(\frac{3}{2}\right)^{\,x} \]