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

To find the exponential function \( f(x) = a^x \) that passes through the point \( \left(2, \frac{1}{4}\right) \), we can substitute \( x = 2 \) and \( f(2) = \frac{1}{4} \) into the equation: \[ \frac{1}{4} = a^2. \] To solve for \( a \), we recognize that \( \frac{1}{4} = \left(\frac{1}{2}\right)^2 \). Thus, we can deduce that \( a = \frac{1}{2} \). So, the function is \( f(x) = \left(\frac{1}{2}\right)^x \). To confirm, when \( x = 2 \), \( f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). Voila! Did you know the concept of exponential functions has been around since the 17th century, with mathematicians like John Napier laying the groundwork? Napier introduced logarithms, connecting multiplication and division to addition and subtraction! It revolutionized calculations, paving the way for the development of exponential functions used today. In practical terms, exponential functions model everything from population growth to radioactive decay! Say you have 1 bacteria cell that doubles every hour. After 5 hours, you can see how quickly it would explode to 32 cells, thanks to the magic of exponential growth. It's like nature's very own magic trick!