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Trigonometry : Period and Amplitude

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Example Question #1 : Period And Amplitude

What is the amplitude in the graph of the following equation:

$-2y=6sin(2\Theta )$

Possible Answers:

Correct answer:

Explanation:

The general form for a sine equation is:

$y=Asin(B(\Theta -C))+D$

The amplitude of a sine equation is the absolute value of .

Since our equation begins with , we would simplify the equation:

$y=-3(\frac{sin(2\Theta )}{2})$

The absolute value of would be .

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Example Question #1 : Period And Amplitude

What is the amplitude of y = sinx ?

Possible Answers:

Correct answer:

Explanation:

Amplitude describes the distance from the middle of a periodic function to its local maximum. y = sinx covers the range from -1 to 1. Thus, it covers a distance of 2 vertically. Half of this, or 1, gives us the amplitude of the function. It is often helpful to think of the amplitude of a periodic function as its "height".

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Example Question #1 : Period And Amplitude

What is the amplitude of y = 4cosx ?

Possible Answers:

Correct answer:

Explanation:

The amplitude of a function describes its height from the midline to the maximum. The amplitude of the parent function, y = cosx , is 1, since it goes from -1 to 1. In this case our function has been multiplied by 4. Think of the effects this multiplication has on the outputs. In , we get our maximum at x = 0 , and cosx = 1 . Here, we will get 4. The same thing happens for our minimum, at $\pi$ , cosx = -1 . Here, we get -4. Thus, by this analysis, it is clear that the amplitude is 4. In the future, remember that the number preceding the cosine function will always be its amplitude.

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Example Question #2 : Period And Amplitude

What is the period of the function y= sinx ?

Possible Answers:

$2\pi$

$\pi$

$3\pi$

Correct answer:

$2\pi$

Explanation:

By definition, the period of a function is the length of for which it repeats. y = sinx starts at 0, continues to 1, goes back to 0, goes to -1, and then back to 0.

This complete cycle goes from x = 0 to x = $2\pi$ .

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Example Question #4 : Period And Amplitude

What is the period and amplitude of the following trigonometric function?

$y = 9\ cos(6\pi x \ + 7) \ + 12$

Possible Answers:

$Amplitude = 3, \ period = \frac{\pi }{3}$

$Amplitude = 9, \ period = 3$

$Amplitude = 7, \ period = 2\pi$

$Amplitude = 12, \ period = 3$

$Amplitude = 9, \ period = \frac{1}{3}$

Correct answer:

$Amplitude = 9, \ period = \frac{1}{3}$

Explanation:

Recall the form of a sinusoid:

$y = A\sin (Bx + C) + D$ or y = Acos(Bx + C) + D

The important quantities for this question are the amplitude, given by , and period given by $\frac{2\pi }{B}{}$ .

For this problem, amplitude is equal to $A = \mathbf{9}$ and period is $\frac{2\pi }{6\pi } = \mathbf{\frac{1}{3}}{}$ .

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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 #5 : Period And Amplitude

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

Possible Answers:

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

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

$y=3\sin\pi x$

$y=3\sin2x$

None of the above

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 #3 : Period And Amplitude

Which of the given functions has the greatest amplitude?

Possible Answers:

$2\sin(x)$

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

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

$\sin(x+3)$

$\sin(4x)$

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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Trigonometry : Period and Amplitude

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 : Period And Amplitude

Example Question #2 : Period And Amplitude

Example Question #4 : Period And Amplitude

Example Question #1 : Period And Amplitude

Example Question #5 : Period And Amplitude

Example Question #3 : Period And Amplitude

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