Trigonometry : Trigonometric Graphs

Study concepts, example questions & explanations for Trigonometry

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Example Questions

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Example Question #1 : Period And Amplitude

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

Possible Answers:

Correct answer:

Explanation:

The general form for a sine equation is:

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

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

The absolute value of  would be .

Example Question #1 : Period And Amplitude

What is the amplitude of

Possible Answers:

Correct answer:

Explanation:

Amplitude describes the distance from the middle of a periodic function to its local maximum.  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". 

Example Question #1 : Period And Amplitude

What is the amplitude of

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, , 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 , and . Here, we will get 4. The same thing happens for our minimum, at  , . 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.

Example Question #1 : Trigonometric Graphs

What is the period of the function

Possible Answers:

Correct answer:

Explanation:

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

This complete cycle goes from  to  .

Example Question #1 : Period And Amplitude

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

Possible Answers:

Correct answer:

Explanation:

Recall the form of a sinusoid:

  or     

The important quantities for this question are the amplitude, given by , and period given by  .

For this problem, amplitude is equal to  and period is .

 

 

Example Question #1 : Period And Amplitude

What is the period of the following function?

Possible Answers:

Correct answer:

Explanation:

The period of the standard cosine function is .

We can find the period of the given function by dividing  by the coefficient in front of , which is :

.

Example Question #7 : Period And Amplitude

Write the equation of sine graph with amplitude 3 and period of

Possible Answers:

None of the above

Correct answer:

Explanation:

Giving 

,

where

 

and 

 

Then,

,

hence 

.

.

Therefore,

Example Question #2 : Trigonometric Graphs

Which of the given functions has the greatest amplitude?

Possible Answers:

Correct answer:

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 .

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.

Example Question #1 : Phase Shifts

Identify the phase shift of the following equation.

Possible Answers:

Correct answer:

Explanation:

If we use the standard form of a sine function

the phase shift can be calculated by .  Therefore, in our case, our phase shift is

Example Question #1 : Phase Shifts

Which of the following is equivalent to 

Possible Answers:

Correct answer:

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  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 .

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

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