Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


y = 3 cos 60x

Verified step by step guidance

1. Identify the function type: The given function is a trigonometric function, specifically a cosine function, y = 3 cos(60x). The amplitude is 3, and the frequency is 60.
2. Determine the period of the function: The period of a cosine function is given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). Here, \( b = 60 \), so the period is \( \frac{2\pi}{60} = \frac{\pi}{30} \).
3. Set the x-axis viewing window: To capture at least one full period of the function, set the x-axis range to include at least \( 0 \) to \( \frac{\pi}{30} \). A good choice might be slightly wider, such as \( -\frac{\pi}{30} \) to \( \frac{2\pi}{30} \), to see the behavior before and after one period.
4. Set the y-axis viewing window: The amplitude of the function is 3, so the y-values will range from -3 to 3. Set the y-axis range slightly wider, such as -4 to 4, to ensure the peaks and troughs are clearly visible.
5. Use graphing software: Input the function y = 3 cos(60x) into graphing software with the determined viewing window. Adjust the window if necessary to ensure the graph displays the overall behavior of the function, including its periodic nature and amplitude.

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions like cosine are periodic, meaning they repeat their values in regular intervals. The period of a function y = cos(bx) is given by 2π/b. For y = 3 cos 60x, the period is 2π/60, which simplifies to π/30. Understanding the period helps in setting the viewing window to capture the function's repeating pattern.

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function determines the height of its peaks and troughs. In the function y = 3 cos 60x, the amplitude is 3, indicating that the graph oscillates between -3 and 3. This information is crucial for setting the vertical limits of the viewing window to ensure the entire range of the function is visible.

Graphing Software Viewing Window

Choosing an appropriate viewing window in graphing software involves setting the x and y-axis limits to effectively display the function's behavior. For y = 3 cos 60x, the x-axis should cover multiple periods to show the function's periodic nature, while the y-axis should accommodate the amplitude. A well-chosen window reveals key features like symmetry and oscillation.