# Trigonometry : Graphing Sine and Cosine

## Example Questions

### Example Question #1 : Graphing Sine And Cosine

The function shown below has an amplitude of ___________ and a period of _________.

Explanation:

The amplitude is always a positive number and is given by the number in front of the trigonometric function.  In this case, the amplitude is 4.  The period is given by , where b is the number in front of x.  In this case, the period is .

### Example Question #1 : Graphing Sine And Cosine

This is the graph of what function?

Explanation:

The amplitude of the sine function is increased by 3, so this is the coefficient for . The +2 shows that the origin of the function is now at  instead of

### Example Question #1 : Graphing Sine And Cosine

Which of the following graphs does not have a -intercept at

Explanation:

The y-intercept is the value of y when .

Recall that cosine is the  value of the unit circle. Thus, , so it works.

Secant is the reciprocal of cosine, so it also works.

Also recall that . Thus, the only answer which is not equivalent is

### Example Question #1 : Graphing Sine And Cosine

What is an equation for the above function?

Explanation:

The amplitude of a sinusoidal function is  unless amplified by a constant in front of the equation. In this case, the amplitude is , so the front constant is .

The graph moves through the origin, so it is either a sine or a shifted cosine graph.

It repeats once in every , as opposed to the usual , so the period is doubled, the constant next to the variable is .

The only answer in which both the correct amplitude and period is found is:

### Example Question #51 : Trigonometric Functions And Graphs

Let  be a function defined as follows:

.

The 3 in the function above affects what attribute of the graph of ?

Phase shift

Period

Amplitude

Vertical shift

Vertical shift

Explanation:

The period of the function is indicated by the coefficient in front of ; here the period is unchanged.

The amplitude of the function is given by the coefficient in front of the ; here the amplitude is 2.

The phase shift is given by the value being added or subtracted inside the function; here the shift is  units to the right.

The only unexamined attribute of the graph is the vertical shift, so 3 is the vertical shift of the graph.

### Example Question #1 : Graphing Sine And Cosine

Which graph correctly illustrates the given equation?

Explanation:

The simplest way to solve a problem like this is to determine where a particular point on the graph would lie and then compare that to our answer choices. We should first find the y-value when the x-value is equal to zero. We will start by substituting zero in for the x-variable in our equation.

Now that we have calculated the y-value we know that the correct graph must have the following point:

Unfortunately, two of our graph choices include this point; thus, we need to pick a second point.

Let's find the y-value when the x-variable equals the following:

We will begin by substituting this into our original equation.

Now we need to investigate the two remaining choices for the following point:

Unfortunately, both of our remaining graphs have this point as well; therefore, we need to pick another x-value. Suppose the x-variable equals the following:

Now, we must substitute this value into our given equation.

Now, we can look for the graph with the following point:

We have narrowed in on our final answer; thus, the following graph is correct: