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Determine the exact maximum and minimum y-values and their corresponding x-values for one period where x > 0.  (For each answer, use the first occurrence for which x > 0.)   maximum     (x, y) = minimum (x, y) =

Determine the exact maximum and minimum y-values and their corresponding x-values for one period where x > 0.  (For each answer, use the first occurrence for which x > 0.)   maximum     (x, y) = minimum (x, y) =

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Determine the exact maximum and minimum y-values and their corresponding x-values for one period where x > 0.

 (For each answer, use the first occurrence for which x > 0.)

 

maximum     (x, y) =
minimum (x, y) =
**Graphing Two Full Periods of the Function** The function represented is \( f(x) = \cos(5x) \). **Graph Explanation:** - **Axes and Scale:** - The horizontal axis (x-axis) ranges from \( -\frac{2\pi}{5} \) to \( \frac{2\pi}{5} \). - The vertical axis (y-axis) ranges from -2 to 2. - **Function Characteristics:** - The function \( f(x) = \cos(5x) \) is a cosine wave with a higher frequency due to the factor of 5 inside the cosine function. - This frequency change affects the period of the cosine function. The standard period of \( \cos(x) \) is \( 2\pi \), and for \( \cos(5x) \), the period becomes \( \frac{2\pi}{5} \). - The graph shows two complete periods of this function, indicating the repetition of its pattern over this interval. - **Graph Behavior:** - The wave starts at the maximum point at \( y = 1 \), decreases to \( y = -1 \), and then returns to the maximum, completing one cycle. - This cyclical pattern repeats for a second full period within the given domain. Understanding this graph helps in visualizing how the cosine function behaves when its frequency is altered, demonstrating how multiple cycles can fit into a compressed interval on the x-axis.
Transcribed Image Text:**Graphing Two Full Periods of the Function** The function represented is \( f(x) = \cos(5x) \). **Graph Explanation:** - **Axes and Scale:** - The horizontal axis (x-axis) ranges from \( -\frac{2\pi}{5} \) to \( \frac{2\pi}{5} \). - The vertical axis (y-axis) ranges from -2 to 2. - **Function Characteristics:** - The function \( f(x) = \cos(5x) \) is a cosine wave with a higher frequency due to the factor of 5 inside the cosine function. - This frequency change affects the period of the cosine function. The standard period of \( \cos(x) \) is \( 2\pi \), and for \( \cos(5x) \), the period becomes \( \frac{2\pi}{5} \). - The graph shows two complete periods of this function, indicating the repetition of its pattern over this interval. - **Graph Behavior:** - The wave starts at the maximum point at \( y = 1 \), decreases to \( y = -1 \), and then returns to the maximum, completing one cycle. - This cyclical pattern repeats for a second full period within the given domain. Understanding this graph helps in visualizing how the cosine function behaves when its frequency is altered, demonstrating how multiple cycles can fit into a compressed interval on the x-axis.
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