# Precalculus : Graphing the Sine and Cosine Functions

## Example Questions

### Example Question #81 : Pre Calculus

How many -intercepts does the function  have in the domain ?

Explanation:

The period of the sine function is .  The period of our new function is 1.  Each period will have two zeroes and there will be one tacked on at the end when the domain is closed.  There will be -intercepts at

The graph is below.

### Example Question #82 : Pre Calculus

If  and , then which of the following must be true about ?

Explanation:

It is a question of what quadrant is in.

A negative value for secant indicates quadrant II or III. Since secant is the reciprocal of cosine, the measurement includes the x value and the r value with regards to position.To get a negative value for secant or cosine we will need a negative x value and either a positive or negative y value to get the correct r value.

A positive value for cotangent indicates quadrant I or III. Since cotangent is the reciprocal of tangent, the measurement includes the x and y values with regards to the position. To get a cotangent that is positive we will need a positive x value and either a positive or negative y value.

The overlap between these two statements is quadrant III. Therefore, must be in quadrant III.

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

Which of the following functions has a y-intercept of

Explanation:

The y-intercept of a function is found by substituting . When we do this to each, we can determine the y-intercept. Don't forget your unit circle!

Thus, the function with a y-intercept of  is

### Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Find the phase shift of .

Explanation:

In the formula,

.

represents the phase shift.

Plugging in what we know gives us:

.

Simplified, the phase is then .

### Example Question #1 : Find The Phase Shift Of A Sine Or Cosine Function

Describe the phase shift of the following function:

Explanation:

Since  is being added inside the parentheses, there will be a horizontal shift. The goal is to maintain zero within the parentheses so you will shift left  radians.

### Example Question #3 : Find The Phase Shift Of A Sine Or Cosine Function

Which equation would produce this graph?

Explanation:

This is the graph of sine, but shifted to the right units. To reflect this shift, should be subtracted from x.

Thus resulting in

.

### Example Question #4 : Find The Phase Shift Of A Sine Or Cosine Function

Which equation would produce this sine graph?

Explanation:

The graph has an amplitude of 2 but has been shifted down 1:

In terms of the equation, this puts a 2 in front of sin, and -1 at the end.

This makes it easier to see that the graph starts [is at 0] where .

The phase shift is to the right, or

### Example Question #5 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a sine graph with a maximum at  and a minimum at .

Explanation:

To write this equation, it is helpful to sketch a graph:

Indicating the maximum and minimum points, we can see that this graph has been shifted up 1, and it has an amplitude of 2.

The distance from the maximum to the minimum point is half the wavelength. In this case, the wavelength is . That means the full wavelength is , and the frequency is 1.

This sketch shows that the graph starts to the left of the y-axis. To figure out exactly where, subtract from the maximum x-coordinate, :

.

Our equation will be in the form where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This graph has an equation of

.

### Example Question #6 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a cosine graph with a maximum at  and a minimum at .

Explanation:

In order to write this equation, it is helpful to sketch a graph:

The dotted line is at , where the maximum occurs and therefore where the graph starts. This means that the graph is shifted to the right

The distance from the maximum to the minimum is half the entire wavelength. Here it is .

Since half the wavelength is , that means the full wavelength is so the frequency is just 1.

The amplitude is 3 because the graph goes symmetrically from -3 to 3.

The equation will be in the form where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

This equation is

.

### Example Question #7 : Find The Phase Shift Of A Sine Or Cosine Function

Write the equation for a sine function with a maximum at  and a minimum at .

Explanation:

The equation will be in the form where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

To write the equation, it is helpful to sketch a graph:

From plotting the maximum and minimum, we can see that the graph is centered on with an amplitude of 3.

The distance from the maximum to the minimum is half the wavelength. For this graph, this distance is .

This means that the total wavelength is and the frequency is 1.

The graph starts behind the maximum point. To determine this x value, subtract from the x-coordinate of the maximum:

Our equation is:

.