# 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

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:

### Example Question #66 : Trigonometric Functions And Graphs

Let  be a function defined as follows:

.

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

Amplitude

Period

Vertical shift

Phase 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

What is an equation for the above function, enlarged below?

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 #1 : Graphing Sine And Cosine

What is the domain of the sine function? What is the domain of the cosine function?

Domain of sine: all real numbers

Domain of cosine: all real numbers

Domain of sine:

Domain of cosine:

Domain of sine:

Domain of cosine: all real numbers

Domain of sine: all real numbers

Domain of cosine:

Domain of sine: all real numbers

Domain of cosine: all real numbers

Explanation:

Both sine and cosine functions go on infinitely to the left and right when viewed on a graph. For this reason, each of these functions has domains of "all real numbers."

Alternatively, each of these functions ranges between -1 and 1 in the y direction. The incorrect answers all include , which is the range of both the sine and the cosine functions.

### Example Question #69 : Trigonometric Functions And Graphs

Which of the following would correctly translate the function  into ?

Shift  1 unit down

Shift  to the left  units

Shift  1 unit up

Shift  to the left  units

Shift  to the right  units

Shift  to the left  units

Explanation:

The graph of  is shown in red below, and the graph of  is shown in blue below. Because the function is periodic, there are infinitely many transformations that could allow  to translate into , but there is only one answer choice below that is correct, and that is "shift  to the left  units." Per the graph, shifting  to the right  units would also be correct, but that is not an available answer choice.

### Example Question #70 : Trigonometric Functions And Graphs

Which of the following graphs represents the function ?