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Chapter 4: Problem 56

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$

Short Answer

Expert verified
The graph of the function \(y=\cos x +3\) is a standard cosine function, shifted upwards 3 units, oscillating between a maximum of 4 and a minimum of 2 from \(x=0\) to \(x=2\pi\).

Step by step solution

01

Base Shape of Cosine Function

Draw the basic cosine function over one period (0 to \(2\pi\)). The cosine function starts at a peak, drops to a trough and returns to the peak. It has a maximum value of 1 at \(x=0\) and a minimum value of -1 at \(x=\pi\).
02

Apply Vertical Shift

Apply the vertical shift by moving the entire graph upward by 3 units. Do this by adding 3 to the y-coordinate for each x-coordinate. The maximum value of the function is now 4 (because 1+3), and the minimum value is 2 (because -1+3). The function still oscillates between these two values over the period \(0\) to \(2\pi\) as before, but shifted up.
03

Draw Final Graph

Draw the final graph of the vertical shift of the cosine function \(y=\cos x+3\) over one period \(0\) to \(2\pi\). The graph starts at a peak of 4 at \(x=0\), drops to a trough of 2 at \(x=\pi\), and returns to the peak of 4 at \(x=2\pi\)

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Key Concepts

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

Cosine Function
Understanding the cosine function is crucial to graphing periodic functions in trigonometry. The cosine function, denoted as \(y = \cos x\), is a fundamental trigonometric function that describes a smooth, wave-like pattern. It is defined for all real numbers and repeats its values in a predictable manner.The basic shape of the cosine graph is periodic with a **period** of \(2\pi\), meaning it completes one full wave cycle in the interval \(0\) to \(2\pi\). It contains:
  • Peaks at 1 when \(x = 0, \pm2\pi, \pm4\pi, ...\)
  • Troughs at -1 when \(x = \pi, \pm3\pi, \pm5\pi, ...\)
  • The midline, where the value is 0, is crossed at \(x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, ...\)
The graph is symmetric about the y-axis, demonstrating an even function. Recognizing these features helps interpret transformations such as shifts and stretches that occur in more complex trigonometric functions.
Vertical Shift
A vertical shift in a graph involves moving the graph up or down without altering its shape. To apply a vertical shift to a function, you add or subtract a constant from the function's equation.For the function \(y = \cos x + 3\), we apply a vertical shift of 3 units upward. The original cosine function \(y = \cos x\) has its maximum and minimum values at 1 and -1, respectively. When we add 3:
  • The **maximum value** shifts from 1 to 4 (\(1 + 3\)).
  • The **minimum value** shifts from -1 to 2 (\(-1 + 3\)).
The whole graph of \(y = \cos x\) is thus elevated, maintaining its oscillating behavior but now oscillating between the values 4 and 2.Vertical shifts do not affect the period or shape of the wave. The graph continues to repeat itself every \(2\pi\) interval, just at a higher position.
Graphing Cosine
Graphing a cosine function after vertical shifts involves understanding how these transformations affect the original wave. To plot \(y = \cos x + 3\), start by considering the characteristics of \(y = \cos x\). Remember that:
  • The function starts at the maximum (peak) at \(x = 0\).
  • Reaches a minimum (trough) at \(x = \pi\).
  • Returns to a maximum at \(x = 2\pi\).
For \(y = \cos x + 3\):
  • Begin plotting from \(x = 0\) at the shifted maximum of 4.
  • At \(x = \pi\), the function is at the shifted minimum of 2.
  • The cycle concludes with the graph returning to 4 at \(x = 2\pi\).
Plotting this graph over one period from \(0\) to \(2\pi\), you will see a wave oscillating between 4 and 2. Introducing the vertical shift in the graph aids in visualizing transformations and understanding how additional constants affect the function's representation in the coordinate plane.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. The exercises use identities, introduced in Section \(4.2,\) that enable you to rewrite trigonometric expressions so that they contain only sines and cosines: $$\begin{array}{ll} \csc x=\frac{1}{\sin x} & \sec x=\frac{1}{\cos x} \\ \tan x=\frac{\sin x}{\cos x} & \cot x=\frac{\cos x}{\sin x} \end{array}$$ Rewrite each expression by changing to sines and cosines. Then simplify the resulting expression. $$\tan x \csc x \cos x$$

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