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Sketch the graphs of the function f(x) = cos x. y y 2 2 f X 3X -27 2 -2- -2- y y 2- f X -27 - 2 2 -1 -2f -2

Sketch the graphs of the function f(x) = cos x. y y 2 2 f X 3X -27 2 -2- -2- y y 2- f X -27 - 2 2 -1 -2f -2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Title: Understanding the Graphs of the Cosine Function** **Introduction to Cosine Function Graphs** The cosine function, denoted as \( f(x) = \cos x \), is one of the fundamental trigonometric functions. Its graph is a wave-like structure that repeats at regular intervals, specifically every \( 2\pi \). **Visual Representation** The image contains four graphs that sketch the function \( f(x) = \cos x \). Each graph showcases a segment of the periodic cosine wave, demonstrating a section of its continuous cycle. **Graph Details** 1. **First Graph (Top Left):** - **Interval:** \(-\pi\) to \(\pi\) - The curve begins at \( x = -\pi \) with a value of \(-1\), peaks at \( x = 0 \) with a value of 1, and descends back to \(-1\) at \( x = \pi \). - The curve passes through the midline at \( y = 0 \) halfway between the peaks and troughs. 2. **Second Graph (Top Right):** - **Interval:** \(-2\pi\) to \(2\pi\) - The cosine wave passes through the points where \( x = -3\pi/2, -\pi/2, \pi/2, 3\pi/2 \) crossing the midline at \( y = 0 \). - Peaks at \( x = -2\pi, 0, 2\pi \) where the value is 1, and troughs at \( x = -\pi, \pi \) where the value is \(-1\). 3. **Third Graph (Bottom Left):** - **Interval:** \(-2\pi\) to \(2\pi\) - Similar to the top right, highlighting the repetitive nature of the cosine function with peaks at integral multiples of \( \pi \). 4. **Fourth Graph (Bottom Right):** - **Interval:** \(-\pi\) to \(\pi\) - Identical to the first graph, reinforcing the idea of periodicity with the cosine function creating the same pattern every \( 2\pi \). **Characteristics Highlighted:** - **Amplitude:** The maximum value (height) is 1, and the minimum value is
Transcribed Image Text:**Title: Understanding the Graphs of the Cosine Function** **Introduction to Cosine Function Graphs** The cosine function, denoted as \( f(x) = \cos x \), is one of the fundamental trigonometric functions. Its graph is a wave-like structure that repeats at regular intervals, specifically every \( 2\pi \). **Visual Representation** The image contains four graphs that sketch the function \( f(x) = \cos x \). Each graph showcases a segment of the periodic cosine wave, demonstrating a section of its continuous cycle. **Graph Details** 1. **First Graph (Top Left):** - **Interval:** \(-\pi\) to \(\pi\) - The curve begins at \( x = -\pi \) with a value of \(-1\), peaks at \( x = 0 \) with a value of 1, and descends back to \(-1\) at \( x = \pi \). - The curve passes through the midline at \( y = 0 \) halfway between the peaks and troughs. 2. **Second Graph (Top Right):** - **Interval:** \(-2\pi\) to \(2\pi\) - The cosine wave passes through the points where \( x = -3\pi/2, -\pi/2, \pi/2, 3\pi/2 \) crossing the midline at \( y = 0 \). - Peaks at \( x = -2\pi, 0, 2\pi \) where the value is 1, and troughs at \( x = -\pi, \pi \) where the value is \(-1\). 3. **Third Graph (Bottom Left):** - **Interval:** \(-2\pi\) to \(2\pi\) - Similar to the top right, highlighting the repetitive nature of the cosine function with peaks at integral multiples of \( \pi \). 4. **Fourth Graph (Bottom Right):** - **Interval:** \(-\pi\) to \(\pi\) - Identical to the first graph, reinforcing the idea of periodicity with the cosine function creating the same pattern every \( 2\pi \). **Characteristics Highlighted:** - **Amplitude:** The maximum value (height) is 1, and the minimum value is
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