Graph each function. $\text{ƒ}\left(x\right)=4^x$

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant and 'x' is the variable. These functions exhibit rapid growth or decay, depending on the base 'a'. In this case, f(x) = 4^x represents an exponential function with a base of 4, which means that as 'x' increases, the value of f(x) increases exponentially.

Graphing Exponential Functions

Graphing exponential functions involves plotting points based on the function's values for various 'x' inputs. The graph of f(x) = 4^x will show a curve that rises steeply as 'x' becomes positive and approaches zero as 'x' becomes negative. Understanding how to identify key points, such as the y-intercept (0,1) when x=0, is essential for accurately representing the function on a coordinate plane.

Transformations of Functions

Transformations of functions refer to changes made to the basic function that affect its graph, such as shifts, stretches, or reflections. For the function f(x) = 4^x, transformations can include vertical shifts (adding or subtracting a constant) or horizontal shifts (changing the input variable). Recognizing how these transformations impact the graph is crucial for understanding variations of the exponential function.