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Trigonometry : Trigonometric Functions and Graphs

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Example Questions

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Example Question #1 : Period And Amplitude

What is the period of the following function?

$f(x)=4\cos\left(\frac{2\pi x}{5}+\frac{\pi}{4}\right)-2$

Possible Answers:

$\frac{2\pi}{5}$

$5\pi$

$\frac{5}{2}$

Correct answer:

Explanation:

The period of the standard cosine function is $2\pi$ .

We can find the period of the given function by dividing $2\pi$ by the coefficient in front of , which is $\frac{2\pi}{5}$ :

$P=\frac{2\pi}{\frac{2\pi}{5}}=2\pi\frac{5}{2\pi}=5$ .

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Example Question #1 : Period And Amplitude

Write the equation of sine graph with amplitude 3 and period of $\pi$ .

Possible Answers:

$y=3\sin\pi x$

$y=3\sin\frac{1}{2}x$

$y=3\sin2x$

None of the above

$y=\frac{1}{3}\sin2x$

Correct answer:

$y=3\sin2x$

Explanation:

Giving

$y=\alpha \sin \beta x$ ,

where

$\alpha=Amplitude$

and

$Period =\frac{2\pi}{\beta}$

Then,

$\alpha=3$

$\pi=\frac{2\pi}{\beta}$ ,

hence

$period=\pi$ .

$\beta=2$ .

Therefore,

$y=3\sin2x$

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Example Question #1 : Find The Amplitude Of A Sine Or Cosine Function

Which of the given functions has the greatest amplitude?

Possible Answers:

$\sin(x+3)$

$2\sin(x)$

$\sin(4x)$

$\sin\frac{2\pi }{3}(x+2)$

$\frac{1}{2}\sin(4x)$

Correct answer:

$2\sin(x)$

Explanation:

The amplitude of a function is the amount by which the graph of the function travels above and below its midline. When graphing a sine function, the value of the amplitude is equivalent to the value of the coefficient of the sine. Similarly, the coefficient associated with the x-value is related to the function's period. The largest coefficient associated with the sine in the provided functions is 2; therefore the correct answer is $2\sin(x)$ .

The amplitude is dictated by the coefficient of the trigonometric function. In this case, all of the other functions have a coefficient of one or one-half.

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Example Question #51 : Trigonometric Functions And Graphs

Identify the phase shift of the following equation.

$y=-3sin(\frac{x}{2}+\frac{\pi}{4})+1$

Possible Answers:

$\frac{\pi}{2}$

$\frac{\pi}{4}$

$\frac{1}{2}$

Correct answer:

$\frac{\pi}{2}$

Explanation:

If we use the standard form of a sine function

y=Asin(Bx-C)+D

the phase shift can be calculated by $\frac{C}{B}$ . Therefore, in our case, our phase shift is

$\frac\frac{}{}{}{}$ $\frac{\frac{\pi}{4}}{\frac{1}{2}}=\frac{\pi}{4}\cdot2=\frac{\pi}{2}$

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Example Question #2 : Phase Shifts

Which of the following is equivalent to $2\sin(2x-3\pi)$

Possible Answers:

$-2\cos(2x-\frac{\pi}{4})$

$2\sin(2x)$

$-2\cos(2(x-\frac{\pi}{4}))$

$2\cos(x)$

$-2\cos(\frac{x}{2}-\frac{\pi}{2})$

Correct answer:

$-2\cos(2(x-\frac{\pi}{4}))$

Explanation:

The first zero can be found by plugging 3π/2 for x, and noting that it is a double period function, the zeros are every π/2, count three back and there is a zero at zero, going down.

A more succinct form for this answer is $-2\sin(2x)$ but that was not one of the options, so a shifted cosine must be the answer.

The first positive peak is at π/4 at -1, so the cosine function will be shifted π/4 to the right and multiplied by -1. The period and amplitude still 2, so the answer becomes $-2\cos(2(x-\frac{\pi}{4}))$ .

To check, plug in π/4 for x and it will come out to -2.

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Example Question #11 : Trigonometric Graphs

Which of the following is the correct definition of a phase shift?

Possible Answers:

The distance a function is shifted vertically from the general position

The distance a function is shifted horizontally from the general position

A measure of the length of a function between vertical asymptotes

The distance a function is shifted diagonally from the general position

Correct answer:

The distance a function is shifted horizontally from the general position

Explanation:

Take the function cos(x) for example. The graph for cos(x) is

If we were to change the function to $cos(x + \pi)$ , our phase shift is $-\pi$ . This means we need to shift our entire graph $\pi$ units to the left.

Our new graph $cos(x + \pi)$ is the following

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Example Question #12 : Trigonometric Graphs

Consider the function $4sec(5(x+\frac{\pi}{3})) +10$ . What is the phase shift of this function?

Possible Answers:

$-\frac{\pi}{3}$

$\frac{\pi}{3}$

Correct answer:

$-\frac{\pi}{3}$

Explanation:

The general form for the secant transformation equation is D + Asec(B(x - C)) . represents the phase shift of the function. When considering $4sec(5(x+\frac{\pi}{3})) +10$ we see that $C = -\frac{\pi}{3}$ . So our phase shift is $-\frac{\pi}{3}$ and we would shift this function $\frac{\pi}{3}$ units to the left of the original secant function’s graph.

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Example Question #13 : Trigonometric Graphs

True or False: If the function sin(x) has a phase shift of $n2\pi$ , then the graph will not be changed.

Possible Answers:

False

True

Correct answer:

True

Explanation:

This is true because the graph sin(x) has a period of $2\pi$ , meaning it repeats itself every $2\pi$ units. So if sin(x) has a phase shift of any multiple of , then it will just overlay the original graph. This is shown below. In orange is the graph of sin(x) and in purple is the graph of $sin(x + 4\pi)$ .

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Example Question #14 : Trigonometric Graphs

Which of the following is the graph of tan(x) with a phase shift of $\frac{\pi}{2}$ ?

Possible Answers:

Screen shot 2020 08 27 at 2.36.53 pm

Screen shot 2020 08 27 at 2.35.10 pm

Screen shot 2020 08 27 at 2.36.46 pm

Screen shot 2020 08 27 at 2.35.20 pm

Correct answer:

Screen shot 2020 08 27 at 2.35.20 pm

Explanation:

Start this problem by graphing the function of tangent.

Screen shot 2020 08 27 at 2.35.10 pm

Now we need to shift this graph $\frac{\pi}{2}$ to the right.

Screen shot 2020 08 27 at 2.35.16 pm

This gives us our answer

Screen shot 2020 08 27 at 2.35.20 pm

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Example Question #1 : Phase Shifts

True or False: The function $csc(x + \pi)$ has a phase shift of $\pi$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

The form of the general cosecant function is D + Acsc(B(x - C)) . So if we have $csc(x + \pi)$ then , which represents the phase shift, is equal to $-\pi$ . This gives us a phase shift of $-\pi$ .

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Trigonometry : Trigonometric Functions and Graphs

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Period And Amplitude

Example Question #1 : Period And Amplitude

Example Question #1 : Find The Amplitude Of A Sine Or Cosine Function

Example Question #51 : Trigonometric Functions And Graphs

Example Question #2 : Phase Shifts

Example Question #11 : Trigonometric Graphs

Example Question #12 : Trigonometric Graphs

Example Question #13 : Trigonometric Graphs

Example Question #14 : Trigonometric Graphs

Example Question #1 : Phase Shifts

All Trigonometry Resources

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