# Trigonometry : Trigonometric Graphs

## Example Questions

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### Example Question #1 : Period And Amplitude

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

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 #2 : Period And Amplitude

What is the amplitude of

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 #3 : Period And Amplitude

What is the amplitude of

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 : Period And Amplitude

What is the period of the function

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 #5 : Period And Amplitude

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

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 #6 : Period And Amplitude

What is the period of the following function?

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

None of the above

Explanation:

Giving

,

where

and

Then,

,

hence

.

.

Therefore,

### Example Question #8 : Period And Amplitude

Which of the given functions has the greatest amplitude?

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.

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 #2 : Phase Shifts

Which of the following is equivalent to