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

Key Concepts

Exponential Functions

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 are characterized by their rapid growth or decay, depending on the value of 'b'. Understanding the general shape and behavior of exponential functions is crucial for analyzing their graphs.

Graphing Exponential Functions

Graphing exponential functions involves plotting points based on the function's equation and understanding key features such as intercepts, asymptotes, and the direction of growth or decay. The graph typically approaches a horizontal asymptote, which is often the x-axis, and can cross the y-axis at the point (0, a). Identifying these features helps in accurately representing the function visually.

Finding the Equation from Points

To find the equation of an exponential function from given points on its graph, one can use the coordinates of these points to set up a system of equations. By substituting the x-values into the general form of the exponential function, you can solve for the constants 'a' and 'b'. This process often involves logarithmic calculations to isolate the base and determine the function's specific characteristics.