# Precalculus Examples

Step 1
Find the asymptotes.
Step 1.1
For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the tangent function, , for equal to to find where the vertical asymptote occurs for .
Step 1.2
Set the inside of the tangent function equal to .
Step 1.3
The basic period for will occur at , where and are vertical asymptotes.
Step 1.4
Find the period to find where the vertical asymptotes exist.
Step 1.4.1
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.4.2
Divide by .
Step 1.5
The vertical asymptotes for occur at , , and every , where is an integer.
Step 1.6
There are only vertical asymptotes for tangent and cotangent functions.
Vertical Asymptotes: for any integer
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: for any integer
No Horizontal Asymptotes
No Oblique Asymptotes
Step 2
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 3
Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
Step 4
Find the period of .
Step 4.1
The period of the function can be calculated using .
Step 4.2
Replace with in the formula for period.
Step 4.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.4
Divide by .
Step 5
Find the phase shift using the formula .
Step 5.1
The phase shift of the function can be calculated from .
Phase Shift:
Step 5.2
Replace the values of and in the equation for phase shift.
Phase Shift:
Step 5.3
Divide by .
Phase Shift:
Phase Shift:
Step 6
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: None
Vertical Shift: None
Step 7
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: for any integer
Amplitude: None
Period:
Phase Shift: None
Vertical Shift: None
Step 8