Try It! Write Exponential Functions 3. Write an exponential function for each set of points. a. \( (1,12),(2,48),(3,192) \), and \( (4,768) \)

To find the exponential function that fits the points \( (1,12), (2,48), (3,192), \) and \( (4,768) \), we can start by observing the pattern in the y-values. Notably, each y-value seems to multiply by a consistent factor as x increases. Calculating the ratios: - From \( 12 \) to \( 48 \): \( 48/12 = 4 \) - From \( 48 \) to \( 192 \): \( 192/48 = 4 \) - From \( 192 \) to \( 768 \): \( 768/192 = 4 \) This consistent ratio indicates that the function likely has the form \( y = a \cdot b^x \), where \( b = 4 \). To find \( a \), we can use one of the points. Plugging in point \( (1,12) \): \[ 12 = a \cdot 4^1 \] \[ 12 = 4a \] \[ a = 3 \] Thus, the exponential function is: \[ y = 3 \cdot 4^x \] --- Now, let’s verify this function with the other points: - For \( x = 2 \): \( y = 3 \cdot 4^2 = 3 \cdot 16 = 48 \) (checks out) - For \( x = 3 \): \( y = 3 \cdot 4^3 = 3 \cdot 64 = 192 \) (checks out) - For \( x = 4 \): \( y = 3 \cdot 4^4 = 3 \cdot 256 = 768 \) (checks out) It confirms that our function is indeed correct!