Which equation represents an exponential function that passes through the point \( (2,80) \) ? \( f(x)=4(x)^{5} \) \( f(x)=5(x)^{4} \) \( f(x)=4(5)^{x} \) \( f(x)=5(4)^{x} \)

To find an exponential function that passes through the point \( (2,80) \), we can evaluate each equation at \( x = 2 \). For \( f(x) = 4(5)^{x} \): \[ f(2) = 4(5^2) = 4(25) = 100 \] (Not the right point.) For \( f(x) = 5(4)^{x} \): \[ f(2) = 5(4^2) = 5(16) = 80 \] (This one works!) Thus, the correct equation is \( f(x) = 5(4)^{x} \). To further clarify how to identify the correct type of function, remember that exponential functions have a form resembling \( f(x) = a(b)^x \), where \( a \) is a constant and \( b \) is a positive base other than 1. In our case, \( a = 5 \) and \( b = 4 \) create an exponential function that indeed fits through the point provided. Next, when working with exponential functions, a common mistake is conflating them with polynomial functions, such as linear ones or quadratics. Exponential functions grow much more quickly than polynomials and involve constants raised to a variable power, so remember to check whether your function's variable is in the exponent to identify it correctly!