In Exercises 1–4, the graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 4^x, g(x) = 4^-x, h(x) = -4^(-x), r(x) = -4^(-x)+3 3.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in 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 whether the base is greater than or less than one. Understanding their general shape and behavior is crucial for identifying specific functions from their graphs.

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity. For exponential functions, this often indicates the value that the function approaches but never reaches. In the given graph, the horizontal asymptote at y=2 suggests that as x increases or decreases, the function stabilizes around this value.

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant to an exponential function can shift its graph vertically. In the context of the question, recognizing how transformations affect the graph helps in selecting the correct function that matches the given graph.