In Exercises 61–64, give the equation of each exponential function whose graph is shown.

Exponential functions are mathematical expressions of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on the value of 'b'. Understanding their general form is crucial for identifying specific exponential functions from their graphs.

To derive the equation of an exponential function from its graph, it is essential to analyze key points on the graph. For instance, the points (0,2), (1,4), and (2,8) indicate specific values of the function at those x-coordinates. These points can be used to set up equations that help determine the constants 'a' and 'b' in the exponential function.

In an exponential function, the initial value 'a' is the output when x = 0, while the base 'b' can be found by examining the ratio of outputs at consecutive x-values. For example, if f(0) = 2 and f(1) = 4, the base can be calculated as b = f(1)/f(0) = 4/2 = 2. This process is vital for constructing the correct equation of the exponential function.