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Chapter 5: Problem 68

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=5 \cos (\pi-2 x)+6$$

Short Answer

Expert verified
The amplitude of the function \(y=5 \cos (\pi-2 x)+6\) is 5 and the period is \(\pi\).

Step by step solution

01

Identify the amplitude

The amplitude is the absolute value of the coefficient on the cosine function, which in this case is 5. Therefore, the amplitude of the function is 5.
02

Identify the period

The period of the cosine function is calculated as \(\frac{2\pi}{|B|}\). Here, \(B = -2\), so the period is \(\frac{2\pi}{2}\), which simplifies to \(\pi\). Therefore, the period of the function is \(\pi\).
03

Graphing the function

Using a graphing utility (like a graphing calculator, a computer program, or an online tool), the function \(y=5 \cos (\pi-2 x)+6\) can be graphed. Remember to include two full periods in the graph. The amplitude and period identified in the previous steps can help in setting up the appropriate scale on the y-axis and x-axis respectively.

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Key Concepts

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

Amplitude of Cosine Function
When working with trigonometric functions such as cosine, understanding the amplitude is crucial. The amplitude of a cosine function determines the height of the wave from its midline. In simple terms, it is how far the highest point of the wave is from the central line of the graph.

For the function \( y = 5 \cos(\pi - 2x) + 6 \), the amplitude is represented by the absolute value of the coefficient in front of the cosine term. Here, this coefficient is 5, so the amplitude is \( |5| = 5 \).

This means that the maximum value of the function above the midline is 5 units, and the minimum value below the midline is also 5 units. The midline itself is shifted up by 6 units due to the plus 6 added to the original cosine function.
  • Amplitude gives the vertical stretch of the graph.
  • A larger amplitude means taller waves; a smaller amplitude means shorter waves.
Period of Cosine Function
The period of a cosine function tells us the horizontal length required for the function to complete one full cycle on the graph. In simpler terms, it is the distance between two consecutive peaks or troughs of the wave.

For the equation \( y = 5 \cos(\pi - 2x) + 6 \), the period is determined by the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \). Here, \( B = -2 \). Consequently, the period is \( \frac{2\pi}{|-2|} = \pi \). This result signifies that a complete wave cycle occurs every \( \pi \) units along the x-axis.
  • A smaller period results in a more frequent recurring wave pattern.
  • A larger period results in a less frequent wave pattern.
Graphing Utilities
Graphing utilities are incredibly useful tools when visualizing complex trigonometric functions. They allow us to quickly and accurately graph functions to understand their key features such as amplitude and period.

With functions like \( y = 5 \cos(\pi - 2x) + 6 \), a graphing calculator or software can create precise graphs. By setting an appropriate scale based on calculated amplitude and period, the utility helps visualize two full cycles of the function efficiently.
  • Graphing tools can adjust the scale to account for amplitude, making it easier to see the graph's full range.
  • They adjust the x-axis scale for the period, ensuring two full wave cycles are visible when requested.
Remember to input values accurately and double-check setup settings to ensure the graph represents the true mathematical behavior of the function.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. $$\cos \theta=-\sqrt{1-\sin ^{2} \theta} \text { for } 90^{\circ} < \theta < 180^{\circ}$$

Irrigation Engineering The cross sections of an irrigation canal are isosceles trapezoids, where the lengths of three of the sides are 8 feet (see figure). The objective is to find the angle \(\theta\) that maximizes the area of the cross sections. [Hint: The area of a trapezoid is given by \(\left.(h / 2)\left(b_{1}+b_{2}\right) .\right]\) (a) Complete seven rows of the table. $$\begin{array}{|c|c|c|c|} \hline \text {Base } I & \text {Base 2} & \text {Altitude} & \text {Area} \\ \hline 8 & 8+16 \cos 10^{\circ} & 8 \sin 10^{\circ} & 22.06 \\ \hline 8 & 8+16 \cos 20^{\circ} & 8 \sin 20^{\circ} & 42.46 \\ \hline \end{array}$$ (b) Use the table feature of a graphing utility to generate additional rows of the table. Use the table to estimate the maximum cross-sectional area. (c) Write the area \(A\) as a function of \(\theta.\) (d) Use the graphing utility to graph the function. Use the graph to estimate the maximum cross-sectional area. How does your estimate compare with that in part (b)?

True or False Determine whether the statement is true or false. Justify your answer. The tangent function can be used to model harmonic motion.

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sec \theta=3\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=7 \pi / 2$$
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