Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.


Graph the function f (x) = sin³ x.

Verified step by step guidance

1. Understand the function: The given function is \( f(x) = \sin^3(x) \), which can also be written as \( f(x) = (\sin(x))^3 \). This means the sine function is cubed, and we need to analyze its behavior.
2. Identify the key features of the function: The sine function \( \sin(x) \) oscillates between -1 and 1, so \( \sin^3(x) \) will also oscillate between -1 and 1. However, the cubing operation will affect the shape of the graph, making the negative values steeper and the positive values slightly flattened.
3. Choose an appropriate viewing window: Since \( \sin(x) \) is periodic with a period of \( 2\pi \), select a viewing window that includes at least one full period, such as \( x \in [-2\pi, 2\pi] \). For the y-axis, set \( y \in [-1, 1] \) to capture the range of the function.
4. Use graphing software: Input the function \( f(x) = \sin^3(x) \) into the graphing software. Adjust the viewing window to \( x \in [-2\pi, 2\pi] \) and \( y \in [-1, 1] \) to ensure the key features of the graph are visible.
5. Analyze the graph: Observe the behavior of the function. Note that the graph will have the same zeros as \( \sin(x) \) (at \( x = n\pi \), where \( n \) is an integer), and the peaks and troughs will be modified due to the cubing operation. The graph will also exhibit odd symmetry, meaning \( f(-x) = -f(x) \).

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visually represent the relationship between the input (x-values) and output (f(x)-values). Understanding how to graph functions is essential for analyzing their behavior, identifying key features such as intercepts, maxima, minima, and periodicity.

Key Features of Functions

Key features of functions include critical points, intercepts, and asymptotes, which provide insight into the function's behavior. For periodic functions like sine, understanding amplitude, period, and phase shift is crucial for accurately representing the function's shape and characteristics on a graph.

Viewing Windows in Graphing

A viewing window in graphing software defines the range of x and y values displayed on the graph. Selecting an appropriate viewing window is vital to ensure that all significant features of the function are visible, allowing for a comprehensive analysis of its behavior over the specified interval.