Problem 14 Plot the graph of the function \... [FREE SOLUTION]

Chapter 0: Problem 14

Plot the graph of the function $f$ in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{1}{2} \sin 2 x+\cos x $$

Short Answer

Expert verified

Plot the graph of the function $f(x)=\frac{1}{2} \sin {2x}+\cos{x}$ using a suitable viewing window, such as the X-axis range $0 \le x \le 2\pi$ and the Y-axis range $-2 \le y \le 2$. This can be done using a graphing calculator or graphing software, showcasing the combined sine and cosine wave patterns with a period of $2\pi$ and combined maximum amplitude of $\frac{3}{2}$.

Step by step solution

Identify the Trigonometric Components and their Amplitudes and Periods

The given function is $f(x)=\frac{1}{2} \sin{2x}+\cos{x}$. It has two trigonometric components: - $\frac{1}{2} \sin{2x}$: the amplitude is $\frac{1}{2}$, and the period is $\frac{2\pi}{2}= \pi$ - $\cos{x}$: the amplitude is 1, and the period is $2\pi$

Determine the Combined Amplitude and Period

The combined amplitude can be found by adding the individual amplitudes at each point and observing the minimum and maximum values the function achieves. The maximum amplitude for $\frac{1}{2} \sin{2x}$ is $\frac{1}{2}$, and the maximum amplitude for $\cos{x}$ is 1. Adding these gives the combined maximum amplitude of $\frac{3}{2}$. The combined period can be found by taking the least common multiple (LCM) of the individual periods. For $\pi$ and $2\pi$, their LCM is $2\pi$.

Choose an Appropriate Viewing Window

We need to choose a viewing window that covers the combined period and amplitude of the function. The X-axis range should cover one or more periods of the function to show its behavior, and the Y-axis range should cover the maximum and minimum amplitudes to display the graph's peaks and valleys. A suitable viewing window for this function would be as follows: - X-axis range: $0 \le x \le 2\pi$ (covers one period of the function) - Y-axis range: $-2 \le y \le 2$ (covers the amplitudes, with a little extra room)

Plot the Graph

Using the selected viewing window, plot the graph of the function $f(x)=\frac{1}{2} \sin {2x}+\cos{x}$ with the X-axis range $0 \le x \le 2\pi$ and the Y-axis range $-2 \le y \le 2$. This step can be done using a graphing calculator or graphing software. Input the function and adjust the window settings to match the chosen viewing window. The graph will have a combination of sine and cosine wave patterns, with a period of $2\pi$ and combined maximum amplitude of $\frac{3}{2}$.

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91影视

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{1}{2} \sin 2 x+\cos x $$

Short Answer

Step by step solution

Identify the Trigonometric Components and their Amplitudes and Periods

Determine the Combined Amplitude and Period

Choose an Appropriate Viewing Window

Plot the Graph

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