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Chapter 6: Problem 56

Exer. 55-62: Use the graph of a trigonometric function to sketch the graph of the equation without plotting points. $$ y=3+\cos x $$

Short Answer

Expert verified
Sketch the graph by vertically shifting \( y = \cos x \) up by 3 units.

Step by step solution

01

Understand the Parent Function

The given function is related to the parent function of cosine, which is \( y = \cos x \). The graph of \( y = \cos x \) has a range of values from -1 to 1, and a period of \( 2\pi \). It typically starts at the maximum value of 1 at \( x = 0 \), decreases to -1, and then returns to 1 after one complete cycle.
02

Apply Vertical Shift

The given function is \( y = 3 + \cos x \). This equation indicates a vertical shift of the cosine function. The entire graph of \( y = \cos x \) is shifted up by 3 units. Therefore, every point on the cosine graph will have 3 added to its y-coordinate: the maximum goes from 1 to 4, the minimum goes from -1 to 2, and the midline shifts from 0 to 3.
03

Sketch the Shifted Graph

Start by drawing the basic shape of the cosine function: a wave starting at its highest point. Since there's no change in amplitude or period, just move the whole graph upwards by 3 units. The midline of the graph is now at \( y = 3 \), with peaks at \( y = 4 \) and troughs at \( y = 2 \). Ensure the wave completes one period over \( 2\pi \) on the x-axis, keeping the same frequency and pattern as the original cosine wave.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function, expressed as \( y = \cos x \), serves as one of the fundamental components in trigonometry. It embodies a simple wave-like shape with specific characteristics. Let's explore these:
  • Periodic Nature: The cosine function completes a full cycle every \(2\pi\) radians. This full cycle repeats the pattern of the wave, making it predictable.
  • Range and Domain: The range of \( y = \cos x \) is from -1 to 1, implying that it swings between these values. Its domain, however, spans all real numbers \( x \).
  • Wave Characteristics: Typically, the wave starts at its highest point (value of 1) when \( x = 0 \), then descends to -1, and climbs back to 1. This produces the familiar up-and-down wave pattern.
The cosine function is a vital tool in analyzing oscillations, rotations, and other wave-related phenomena.
Vertical Shift
A vertical shift involves moving the entire graph of a function up or down without altering its shape. For the function given as \( y = 3 + \cos x \), the cosine graph undergoes a vertical shift.
  • Understanding the Shift: Adding 3 to \( \cos x \) in the equation results in each point on the graph being raised by 3 units. Consequently, the usual range of the graph is adjusted.
  • New Midline, Maximum, and Minimum: Originally, the midline (equidistant line from peaks and troughs) of \( y = \cos x \) is 0. With the shift, it now becomes 3. Therefore, the maximum rises from 1 to 4, and the minimum climbs from -1 to 2.
This adjustment is purely vertical, affecting only the vertical placement of the function while maintaining the original wave's shape and periodicity.
Graph Sketching
Graphing a function involves visually representing its behavior across specified intervals. To sketch \( y = 3 + \cos x \), consider the vertical shift and the fundamental properties of the cosine wave.
  • Start Simple: Begin by drawing the basic cosine wave, knowing it starts at its peak and completes a full cycle over \( 2\pi \).
  • Apply the Shift: Once you've visualized the cosine wave, translate the whole graph 3 units upward. This respects the new midline at \( y = 3 \), maximum at \( y = 4 \), and minimum at \( y = 2 \).
  • Check Key Features: Ensure that the wave extends horizontally over intervals of \( 2\pi \) while maintaining an equal distribution of peaks and valleys around the midline.
Graph sketches serve as insightful tools for understanding the periodic and amplitude-oriented nature of trigonometric functions, including those affected by shifts like in this exercise.

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Most popular questions from this chapter

Exer. 9-16: Let \(P\) be the point on the unit circle \(U\) that corresponds to \(t\). Find the coordinates of \(P\) and the exact values of the trigonometric functions of \(t\), whenever possible. (a) \(3 \pi / 4\) (b) \(-7 \pi / 4\)

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