Precalculus Examples

y = 5 sin (x) - 1

$y = 5 \sin (x) - 1$

Step 1

Use the form

a sin (b x - c) + d

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

a = 5

$a = 5$

b = 1

$b = 1$

c = 0

$c = 0$

d = - 1

$d = - 1$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

5

$5$

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

5 sin (x)

$5 \sin (x)$ .

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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

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.1.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.1.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.2

Find the period of

- 1

$- 1$ .

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Step 3.2.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.2.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.2.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.3

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

2 π

$2 π$

2 π

$2 π$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 4.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{0}{1}

$\frac{0}{1}$

Step 4.3

Divide

0

$0$ by

1

$1$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

5

$5$

Period:

2 π

$2 π$

Phase Shift: None

Vertical Shift:

- 1

$- 1$

Step 6

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