Algebra Examples

y = 8 cos (5 π x + \frac{3 π}{2}) - 9

$y = 8 \cos (5 π x + \frac{3 π}{2}) - 9$

Step 1

Use the form

a cos (b x - c) + d

$a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = 8

$a = 8$

b = 5 π

$b = 5 π$

c = - \frac{3 π}{2}

$c = - \frac{3 π}{2}$

d = - 9

$d = - 9$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

8

$8$

Step 3

Find the period using the formula

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of

8 cos (5 π x + \frac{3 π}{2})

$8 \cos (5 π x + \frac{3 π}{2})$ .

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Step 3.1.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.1.2

Replace

b

$b$ with

5 π

$5 π$ in the formula for period.

\frac{2 π}{| 5 π |}

$\frac{2 π}{| 5 π |}$

Step 3.1.3

5 π

$5 π$ is approximately

15.70796326

$15.70796326$ which is positive so remove the absolute value

\frac{2 π}{5 π}

$\frac{2 π}{5 π}$

Step 3.1.4

Cancel the common factor of

π

$π$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{5 π}$

Step 3.1.4.2

Rewrite the expression.

$\frac{2}{5}$

Step 3.2

Find the period of

$- 9$ .

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Step 3.2.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.2

Replace

$b$ with

$5 π$ in the formula for period.

$\frac{2 π}{| 5 π |}$

Step 3.2.3

$5 π$ is approximately

$15.70796326$ which is positive so remove the absolute value

$\frac{2 π}{5 π}$

Step 3.2.4

Cancel the common factor of

$π$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{5 π}$

Step 3.2.4.2

Rewrite the expression.

$\frac{2}{5}$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{2}{5}$

Step 4

Find the phase shift using the formula

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

$\frac{c}{b}$ .

Phase Shift:

$\frac{c}{b}$

Step 4.2

Replace the values of

$c$ and

$b$ in the equation for phase shift.

Phase Shift:

$\frac{- \frac{3 π}{2}}{5 π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$- \frac{3 π}{2} \cdot \frac{1}{5 π}$

Step 4.4

Cancel the common factor of

$π$ .

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Step 4.4.1

Move the leading negative in

$- \frac{3 π}{2}$ into the numerator.

Phase Shift:

$\frac{- 3 π}{2} \cdot \frac{1}{5 π}$

Step 4.4.2

Factor

$π$ out of

$- 3 π$ .

Phase Shift:

$\frac{π \cdot - 3}{2} \cdot \frac{1}{5 π}$

Step 4.4.3

Factor

$π$ out of

$5 π$ .

Phase Shift:

$\frac{π \cdot - 3}{2} \cdot \frac{1}{π \cdot 5}$

Step 4.4.4

Cancel the common factor.

Phase Shift:

$\frac{π \cdot - 3}{2} \cdot \frac{1}{π \cdot 5}$

Step 4.4.5

Rewrite the expression.

Phase Shift:

$\frac{- 3}{2} \cdot \frac{1}{5}$

Phase Shift:

$\frac{- 3}{2} \cdot \frac{1}{5}$

Step 4.5

Multiply

$\frac{- 3}{2}$ by

$\frac{1}{5}$ .

Phase Shift:

$\frac{- 3}{2 \cdot 5}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Multiply

$2$ by

$5$ .

Phase Shift:

$\frac{- 3}{10}$

Step 4.6.2

Move the negative in front of the fraction.

Phase Shift:

$- \frac{3}{10}$

Phase Shift:

$- \frac{3}{10}$

Phase Shift:

$- \frac{3}{10}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$8$

Period:

$\frac{2}{5}$

Phase Shift:

$- \frac{3}{10}$ (

$\frac{3}{10}$ to the left)

Vertical Shift:

$- 9$

Step 6