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Chapter 4: Problem 34

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=2 \cos 2 x\\\ &g(x)=-\cos 4 x \end{aligned}$$

Short Answer

Expert verified
The graphs for the functions \(f(x)=2 \cos 2x\) and \(g(x)=-\cos 4x\) show different behaviour due to their amplitudes and frequencies. \(f(x)\) completes four waves and ranges from -2 to 2, whereas \(g(x)\) completes eight waves and ranges from -1 to 1, when shown over two full periods from 0 to \(4\pi\).

Step by step solution

01

Plot function \(f\)

Start by plotting the function \(f(x)=2 \cos 2x\). The regular cosine function has a period from 0 to \(2\pi\), but this function has a frequency of \(2x\), so it will complete two periods in the range from 0 to \(2\pi\). The amplitude is 2, so the function will range from -2 to 2.
02

Plot function \(g\)

Next plot the function \(g(x)=-\cos 4x\). The negative sign flips the graph of the function over the x-axis. This function has a frequency of \(4x\), so it completes four periods in the range from 0 to \(2\pi\). As the amplitude is 1, it will range from -1 to 1.
03

Show two full periods

To meet the requirement of the exercise, it is needed to extend the x-axis of the graph from \(0\) to \(4\pi\) so that two full periods of both functions are visible. This will show that \(f(x)=2 \cos 2x\) completes four waves (or cycles) for \(0 < x < 4\pi\) and \(g(x)=-\cos 4x\) completes eight waves for \(0 < x < 4\pi\).

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Key Concepts

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

Understanding Amplitude
Amplitude refers to the height of the wave from its center line to its peak or trough. It measures how far the graph of a function stretches vertically. Essentially, amplitude tells you the maximum value the trigonometric function will reach.

For a basic cosine function, which is expressed as \(y = \cos x\), the amplitude is 1. This means that the highest point on its graph is 1 and the lowest point is -1.
  • In the function \(f(x) = 2 \cos 2x\), the coefficient 2 in front of the cosine function signifies that the amplitude is 2. Hence, this function stretches or shrinks vertically between -2 and 2.
  • Similarly, for \(g(x) = -\cos 4x\), the amplitude remains 1, as the absolute value of the coefficient of the cosine is 1, although the graph is flipped due to the negative sign.

Understanding amplitude is crucial when graphing trigonometric functions, as it gives a clear picture of the function鈥檚 vertical reach.
Exploring Period of a Function
The period of a trigonometric function is the distance along the x-axis required for the function to complete one full cycle. For the standard cosine function \(y = \cos x\), this period is \(2\pi\).

Here's how the period is determined for a function \(y = \cos Bx\):
  • Calculate the period using \( \frac{2\pi}{B} \).
  • The value of \(B\) is a 'frequency factor'. It indicates the number of cycles the function completes in the regular period of \(2\pi\).
For \(f(x) = 2 \cos 2x\):
  • The frequency factor is 2, meaning it completes 2 cycles from 0 to \(2\pi\).
  • The period is \( \frac{2\pi}{2} = \pi\).

For \(g(x) = -\cos 4x\):
  • The frequency factor is 4, indicating that the function completes 4 cycles within \(2\pi\).
  • The period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).

Grasping the concept of the period is essential for predicting how often the function repeats itself along the x-axis.
Features of the Cosine Function
The cosine function is one of the fundamental trigonometric functions and is depicted as \(y = \cos x\). It has a few distinctive properties that set it apart:
  • Symmetry: Cosine is an even function, meaning \(\cos(-x) = \cos x\). This symmetry around the y-axis gives it a distinctive U-shape.
  • Maximum and Minimum Values: The cosine function fluctuates from 1 to -1 as it goes through its periods, with each full cycle taking \(2\pi\).

When you modify the basic cosine function, like in \(f(x) = 2 \cos 2x\) and \(g(x) = -\cos 4x\), these features are affected:
  • In \(f(x)\), the "2" multiplies the cosine, doubling its standard range and altering its period to \(\pi\).
  • In \(g(x)\), the negative sign flips the graph vertically around the x-axis, while the "4" increases its frequency to 4 cycles every \(2\pi\).

These transformations maintain the inherent waves and periodicity but adjust their amplitudes and periods. Understanding these qualities is crucial for accurately sketching or predicting the behavior of cosine graphs.

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Most popular questions from this chapter

In calculus, it is shown that the area of the region bounded by the graphs of \(y=0\) \(y=1 /\left(x^{2}+1\right), x=a,\) and \(x=b\) is given by Area \(=\arctan b-\arctan a\) (see figure). Find the area for the following values of \(a\) and \(b.\) (a) \(a=0, b=1\) (b) \(a=-1, b=1\) (c) \(a=0, b=3\) (d) \(a=-1, b=3\)

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in seconds) can be modeled by $$h(t)=53+50 \sin \left(\frac{\pi}{10} t-\frac{\pi}{2}\right)$$ (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the amplitude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model.

A 20 -meter line is a tether for a helium-filled balloon. Because of a breeze, the line makes an angle of approximately \(85^{\circ}\) with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write and solve an equation for the height of the balloon. (c) The breeze becomes stronger and the angle the line makes with the ground decreases. How does this affect the triangle you drew in part (a)? (d) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures \(\theta\) $$\begin{array}{|l|l|l|l|l|} \hline \text { Angle, } \boldsymbol{\theta} & 80^{\circ} & 70^{\circ} & 60^{\circ} & 50^{\circ} \\ \hline \text { Height } & & & & \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline \text { Angle, } \theta & 40^{\circ} & 30^{\circ} & 20^{\circ} & 10^{\circ} \\ \hline \text { Height } & & & & \\ \hline \end{array}$$ (e) As \(\theta\) approaches \(0^{\circ},\) how does this affect the height of the balloon? Draw a right triangle to explain your reasoning.

Find two solutions of each equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi) .\) Do not use a calculator. (a) \(\tan \theta=1\) (b) \(\cot \theta=-\sqrt{3}\)

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the interval \(-2 \pi, 2 \pi .\) Use the graphs to find real numbers \(x\) such that \(y_{1}=y_{2}\). $$\begin{aligned} &y_{1}=\cos x\\\ &y_{2}=-1 \end{aligned}$$
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