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Mathematics LibreTexts

6.R: Periodic Functions (Review)

  • Page ID
    18275
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    6.1: Graphs of the Sine and Cosine Functions

    For the exercises 1-8, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

    1) \(f(x)=-3\cos x+3\)

    Answer

    amplitude: \(3\); period: \(2\pi \); midline: \(y=3\);no asymptotes

    6R6.1.1.png

    2) \(f(x)=\dfrac{1}{4}\sin x\)

    3) \(f(x)=3\cos\left ( x+\dfrac{\pi }{6} \right )\)

    Answer

    amplitude: \(3\); period: \(2\pi \); midline: \(y=0\); no asymptotes

    6R6.1.3.png

    4) \(f(x)=-2\sin\left ( x-\dfrac{2\pi }{3} \right )\)

    5) \(f(x)=3\sin\left ( x-\dfrac{\pi }{4} \right )-4\)

    Answer

    amplitude: \(3\); period: \(2\pi \); midline: \(y=-4\); no asymptotes

    6R6.1.5.png

    6) \(f(x)=2\left (\cos\left ( x-\dfrac{4\pi }{3} \right )+1 \right )\)

    7) \(f(x)=6\sin\left ( 3x-\dfrac{\pi }{6} \right )-1\)

    Answer

    amplitude: \(6\); period: \(dfrac{2\pi }{3}\); midline: \(y=-1\); no asymptotes

    6R6.1.7.png

    8) \(f(x)=-100\sin(50x-20)\)

    6.2: Graphs of the Other Trigonometric Functions

    For the exercises 1-4, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

    1) \(f(x)=\tan x-4\)

    Answer

    stretching factor: none; period: \(\pi \); midline: \(y=-4\); asymptotes: \(x=\dfrac{\pi }{2}+\pi k\), where \(k\) is an integer

    6R6.2.1.png

    2) \(f(x)=2\tan \left ( x-\dfrac{\pi }{6} \right )\)

    3) \(f(x)=-3\tan (4x)-2\)

    Answer

    stretching factor: \(3\); period: \(\dfrac{\pi }{4}\); midline: \(y=-2\); asymptotes: \(x=\dfrac{\pi }{8}+\dfrac{\pi }{4}k\), where \(k\) is an integer

    6R6.2.3.png

    4) \(f(x)=0.2\cos(0.1x)+0.3\)

    For the exercises 5-10, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

    5) \(f(x)=\dfrac{1}{3}\sec x\)

    Answer

    amplitude: none; period: \(2\pi \); no phase shift; asymptotes: \(x=\dfrac{\pi }{2}k\), where \(k\) is an integer

    6R6.2.5.png

    6) \(f(x)=3\cot x\)

    7) \(f(x)=4\csc (5x)\)

    Answer

    amplitude: none; period: \(\dfrac{2\pi }{5}\); no phase shift; asymptotes: \(x=\dfrac{\pi }{5}k\), where \(k\) is an integer

    6R6.2.7.png

    8) \(f(x)=8\sec \left (\dfrac{1}{4}x \right )\)

    9) \(f(x)=\dfrac{2}{3}\csc \left (\dfrac{1}{2}x \right )\)

    Answer

    amplitude: none; period: \(4\pi \); no phase shift; asymptotes: \(x=2\pi k\), where \(k\) is an integer

    6R6.2.9.png

    10) \(f(x)=-\csc (2x+\pi)\)

    For the exercises 11-15, use this scenario: The population of a city has risen and fallen over a \(20\)-year interval. Its population may be modeled by the following function: \(y=12,000+8,000\sin(0.628x)\), where the domain is the years since 1980 and the range is the population of the city.

    11) What is the largest and smallest population the city may have?

    Answer

    largest: \(20,000\); smallest: \(4,000\)

    12) Graph the function on the domain of \([0,40]\).

    13) What are the amplitude, period, and phase shift for the function?

    Answer

    amplitude: \(8,000\); period: \(10\); phase shift: \(0\)

    14) Over this domain, when does the population reach \(18,000\)? \(13,000\)?

    15) What is the predicted population in 2007? 2010?

    Answer

    In 2007, the predicted population is \(4,413\). In 2010, the population will be \(11,924\).

    For the exercises 16a-16d, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

    16) Suppose the graph of the displacement function is shown in the Figure below, where the values on the \(x\)-axis represent the time in seconds and the \(y\)-axis represents the displacement in inches.

    6R6.2.16.png

    1. Give the equation that models the vertical displacement of the weight on the spring.
    2. At \(\text{time} = 0\), what is the displacement of the weight?
    Answer

    \(5\) in.

    1. At what time does the displacement from the equilibrium point equal zero?
    2. What is the time required for the weight to return to its initial height of \(5\) inches? In other words, what is the period for the displacement function?
    Answer

    \(10\) seconds

    6.3: Inverse Trigonometric Functions

    For the exercises 1-11, find the exact value without the aid of a calculator.

    1) \(\sin ^{-1}(1)\)

    2) \(\cos ^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )\)

    Answer

    \(\dfrac{\pi }{6}\)

    3) \(\tan ^{-1}(-1)\)

    4) \(\cos ^{-1}\left ( \dfrac{1}{\sqrt{2}} \right )\)

    Answer

    \(\dfrac{\pi }{4}\)

    5) \(\sin ^{-1}\left ( \dfrac{-\sqrt{3}}{2} \right )\)

    6) \(\sin ^{-1}\left (\cos \left (\dfrac{\pi }{6} \right ) \right )\)

    Answer

    \(\dfrac{\pi }{3}\)

    7) \(\cos ^{-1}\left (\tan \left (\dfrac{3\pi }{4} \right ) \right )\)

    8) \(\sin \left (\sec^{-1} \left (\dfrac{3}{5} \right ) \right )\)

    Answer

    No solution

    9) \(\cot \left (\sin^{-1} \left (\dfrac{3}{5} \right ) \right )\)

    10) \(\tan \left (\cos^{-1} \left (\dfrac{5}{13} \right ) \right )\)

    Answer

    \(\dfrac{12}{5}\)

    11) \(\sin \left (\cos^{-1} \left (\dfrac{x}{x+1} \right ) \right )\)

    12) Graph \(f(x)=\cos x\) and \(f(x)=\sec x\) on the interval \([0,2\pi )\) and explain any observations.

    Answer

    The graphs are not symmetrical with respect to the line \(y=x\).They are symmetrical with respect to the \(y\)-axis.

    6R6.3.12.png

    13) Graph \(f(x)=\sin x\) and \(f(x)=\csc x\) and explain any observations.

    14) Graph the function \(f(x)=\dfrac{x}{1}-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}\) on the interval \([-1,1]\) and compare the graph to the graph of \(f(x)=\sin x\) on the same interval. Describe any observations.

    Answer

    The graphs appear to be identical.

    6R6.3.14.png

    Practice Test

    For the exercises 1-13, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

    1) \(f(x)=0.5\sin x\)

    Answer

    amplitude: \(0.5\); period: \(2\pi \);midline \(y=0\)