Trigonometry Examples

y = cos (- 2 x + π) - 1

$y = \cos (- 2 x + π) - 1$

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 = 1

$a = 1$

b = - 2

$b = - 2$

c = - π

$c = - π$

d = - 1

$d = - 1$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

1

$1$

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

cos (- 2 x + π)

$\cos (- 2 x + π)$ .

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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

- 2

$- 2$ in the formula for period.

\frac{2 π}{| - 2 |}

$\frac{2 π}{| - 2 |}$

Step 3.1.3

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

- 2

$- 2$ and

0

$0$ is

2

$2$ .

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 3.1.4

Cancel the common factor of

2

$2$ .

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Step 3.1.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.1.4.2

Divide

$π$ by

$1$ .

$π$

Step 3.2

Find the period of

$- 1$ .

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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

$- 2$ in the formula for period.

$\frac{2 π}{| - 2 |}$

Step 3.2.3

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

$- 2$ and

$0$ is

$2$ .

$\frac{2 π}{2}$

Step 3.2.4

Cancel the common factor of

$2$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.4.2

Divide

$π$ by

$1$ .

$π$

Step 3.3

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

$π$

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:

$\frac{- π}{- 2}$

Step 4.3

Dividing two negative values results in a positive value.

Phase Shift:

$\frac{π}{2}$

Phase Shift:

$\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$π$

Phase Shift:

$\frac{π}{2}$ (

$\frac{π}{2}$ to the right)

Vertical Shift:

$- 1$

Step 6