Question

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.

Accepted Answer

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. 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. 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. 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. 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 $$ f(-x) = -f(x) .$$

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.

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Concetti chiave

Graphing Functions

Key Features of Functions

Viewing Windows in Graphing