# Graph minus two minus two tan(four x minus two pi)

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 .

Calculate .

Move all terms not containing to the right side of the equation.

Add to both sides of the equation.

To write as a fraction with a common denominator, First, multiply by .

Add and .

Add numerators over the common denominator.

Clarify the numerator.

First, multiply by .

Add and .

First, divide each term by and simplify.

Divide each term in by .

The common factor should be canceled of .

Cancel the common factor.

Divide by .

Multiply .

Multiply and .

Multiply by .

Set the inside of the tangent function equal to .

Calculate .

Move all terms not containing to the right side of the equation.

Add to both sides of the equation.

To write as a fraction with a common denominator, multiply by .

Combine and .

Combine the numerators over the common denominator.

Clarify the numerator.

Multiply by .

Add and .

Divide each term by and simplify.

Divide each term in by .

The common factor should be canceled.

Cancel the common factor.

Divide by .

Multiply .

Multiply and .

Multiply by .

The basic period for will occur at , where and are vertical asymptotes.

The absolute value is the distance between a number and zero. The distance between and is .

The vertical asymptotes for occur at , , and every , where is an integer.

Tangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: where is an integer

Reformulate the expression as .

Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Evaluate the period of function using .

Replace with in the formula for period.

The absolute value is the distance between a number and zero. The distance between and is .

The phase shift of the function can be calculated from .

Phase Shift:

Replace the values of and in the equation for phase shift.

Phase Shift:

The common factor should be canceled and .

Factor out of .

Phase Shift:

The common factors should be canceled.

Factor out of .

Phase Shift:

Cancel the common factor.

Phase Shift:

Should be rewritten the expression.

Phase Shift:

Phase Shift:

Phase Shift:

Phase Shift:

Find the vertical shift .

Vertical Shift:

List the properties of the trigonometric function.

Amplitude: None

Period:

Phase Shift: ( to the right)

Vertical Shift:

Use the amplitude to plot the trig function, period, phase shift, vertical shift, and the points.

Vertical Asymptotes: where is an integer

Amplitude: None

Period:

Phase Shift: ( to the right)

Vertical Shift:

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