Score: \( 15 / 100 \) Answered: \( 3 / 17 \) Find the exponential function \( f(x)=a^{x} \) whose graph goes through the point \( \left(2, \frac{1}{4}\right) \). \( a= \) Question 4 .s Question Help: Video Submit Question

To find the exponential function \( f(x) = a^x \) that passes through the point \( (2, \frac{1}{4}) \), start by substituting the x-value and corresponding f(x) value into the equation. This gives us: \[ \frac{1}{4} = a^2 \] Next, we need to solve for \( a \). Recall that \( \frac{1}{4} \) can be rewritten as \( 4^{-1} \) or \( 2^{-2} \). Thus, we have: \[ a^2 = 2^{-2} \] Taking the square root of both sides to solve for \( a \), we find: \[ a = 2^{-1} = \frac{1}{2} \] So the exponential function is \( f(x) = \left(\frac{1}{2}\right)^x \). And voila! You've successfully derived the function. Those tricky points can feel daunting, but once you break them down, they’re just little puzzles waiting for your genius touch! Looking to explore the beauty of exponential growth and decay further? You might want to delve into how birds use the concept of exponential functions to understand migration patterns or even how bacteria grow exponentially in a controlled environment! It’s not just math; it’s all around us!