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 1.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay, depending on whether 'a' is greater than or less than 1. The graph of an exponential function is characterized by a continuous curve that approaches the x-axis but never touches it, known as a horizontal asymptote.

Graph Characteristics

The graph of an exponential function has distinct characteristics, including a y-intercept at (0, a) and a horizontal asymptote along the x-axis. If the base 'a' is greater than 1, the function increases as x increases; if 'a' is between 0 and 1, the function decreases. The steepness of the curve is influenced by the value of 'a', with larger bases resulting in steeper growth.

Transformations of Functions

Transformations of functions involve shifting, reflecting, or stretching the graph of a function. For example, a negative exponent indicates a reflection across the y-axis, while adding a constant shifts the graph vertically. Understanding these transformations is crucial for identifying the correct function that matches a given graph, as they alter the basic shape and position of the exponential function.