# Precalculus Examples

Step 1

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

Step 2

Find the amplitude .

Amplitude:

Step 3

Step 3.1

The period of the function can be calculated using .

Step 3.2

Replace with in the formula for period.

Step 3.3

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

Step 3.4

Cancel the common factor of and .

Step 3.4.1

Factor out of .

Step 3.4.2

Cancel the common factors.

Step 3.4.2.1

Factor out of .

Step 3.4.2.2

Cancel the common factor.

Step 3.4.2.3

Rewrite the expression.

Step 4

Step 4.1

The phase shift of the function can be calculated from .

Phase Shift:

Step 4.2

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

Phase Shift:

Step 4.3

Divide by .

Phase Shift:

Phase Shift:

Step 5

List the properties of the trigonometric function.

Amplitude:

Period:

Phase Shift: None

Vertical Shift: None

Step 6

Step 6.1

Find the point at .

Step 6.1.1

Replace the variable with in the expression.

Step 6.1.2

Simplify the result.

Step 6.1.2.1

Multiply by .

Step 6.1.2.2

The exact value of is .

Step 6.1.2.3

Multiply by .

Step 6.1.2.4

The final answer is .

Step 6.2

Find the point at .

Step 6.2.1

Replace the variable with in the expression.

Step 6.2.2

Simplify the result.

Step 6.2.2.1

Cancel the common factor of .

Step 6.2.2.1.1

Factor out of .

Step 6.2.2.1.2

Cancel the common factor.

Step 6.2.2.1.3

Rewrite the expression.

Step 6.2.2.2

The exact value of is .

Step 6.2.2.3

Multiply by .

Step 6.2.2.4

The final answer is .

Step 6.3

Find the point at .

Step 6.3.1

Replace the variable with in the expression.

Step 6.3.2

Simplify the result.

Step 6.3.2.1

Cancel the common factor of .

Step 6.3.2.1.1

Cancel the common factor.

Step 6.3.2.1.2

Rewrite the expression.

Step 6.3.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Step 6.3.2.3

The exact value of is .

Step 6.3.2.4

Multiply by .

Step 6.3.2.5

The final answer is .

Step 6.4

Find the point at .

Step 6.4.1

Replace the variable with in the expression.

Step 6.4.2

Simplify the result.

Step 6.4.2.1

Cancel the common factor of .

Step 6.4.2.1.1

Factor out of .

Step 6.4.2.1.2

Cancel the common factor.

Step 6.4.2.1.3

Rewrite the expression.

Step 6.4.2.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

Step 6.4.2.3

The exact value of is .

Step 6.4.2.4

Multiply .

Step 6.4.2.4.1

Multiply by .

Step 6.4.2.4.2

Multiply by .

Step 6.4.2.5

The final answer is .

Step 6.5

Find the point at .

Step 6.5.1

Replace the variable with in the expression.

Step 6.5.2

Simplify the result.

Step 6.5.2.1

Cancel the common factor of .

Step 6.5.2.1.1

Factor out of .

Step 6.5.2.1.2

Cancel the common factor.

Step 6.5.2.1.3

Rewrite the expression.

Step 6.5.2.2

Subtract full rotations of until the angle is greater than or equal to and less than .

Step 6.5.2.3

The exact value of is .

Step 6.5.2.4

Multiply by .

Step 6.5.2.5

The final answer is .

Step 6.6

List the points in a table.

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude:

Period:

Phase Shift: None

Vertical Shift: None

Step 8