Write an exponential function for each set of points. a. \( (0,3),(1,12),(2,48),(3,192) \), and \( (4.768) \)

To find an exponential function that fits the given set of points \((0, 3)\), \((1, 12)\), \((2, 48)\), \((3, 192)\), and \((4, 768)\), we can use the general form of an exponential function: \[ y = ab^x \] **Step 1: Determine the value of \(a\)** Using the point \((0, 3)\): \[ y = ab^0 \implies 3 = a \cdot 1 \implies a = 3 \] **Step 2: Determine the value of \(b\)** Using the point \((1, 12)\): \[ y = 3b^1 \implies 12 = 3b \implies b = \frac{12}{3} = 4 \] **Step 3: Formulate the exponential function** Now that we've found \(a = 3\) and \(b = 4\), the exponential function becomes: \[ y = 3 \cdot 4^x \] **Verification:** Let's verify this function with another point, say \((2, 48)\): \[ y = 3 \cdot 4^2 = 3 \cdot 16 = 48 \] The point fits perfectly. Similarly, you can verify the other points: - \((3, 192)\): \(3 \cdot 4^3 = 3 \cdot 64 = 192\) - \((4, 768)\): \(3 \cdot 4^4 = 3 \cdot 256 = 768\) All points satisfy the function. **Final Answer:** \[ y = 3 \times 4^{\,x} \]