Algebra Examples

y = sin (4 π x)

$y = \sin (4 π x)$

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 = 1

$a = 1$

b = 4 π

$b = 4 π$

c = 0

$c = 0$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

1

$1$

Step 3

Find the period of

sin (4 π x)

$\sin (4 π x)$ .

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Step 3.1

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

Replace

b

$b$ with

4 π

$4 π$ in the formula for period.

\frac{2 π}{| 4 π |}

$\frac{2 π}{| 4 π |}$

Step 3.3

4 π

$4 π$ is approximately

12.56637061

$12.56637061$ which is positive so remove the absolute value

\frac{2 π}{4 π}

$\frac{2 π}{4 π}$

Step 3.4

Cancel the common factor of

2

$2$ and

4

$4$ .

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Step 3.4.1

Factor

2

$2$ out of

2 π

$2 π$ .

\frac{2 (π)}{4 π}

$\frac{2 (π)}{4 π}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

2

$2$ out of

4 π

$4 π$ .

\frac{2 (π)}{2 (2 π)}

$\frac{2 (π)}{2 (2 π)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (2 π)}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{2 π}$

Step 3.5

Cancel the common factor of

$π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{2 π}$

Step 3.5.2

Rewrite the expression.

$\frac{1}{2}$

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:

$\frac{0}{4 π}$

Step 4.3

Cancel the common factor of

$0$ and

$4$ .

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Step 4.3.1

Factor

$4$ out of

$0$ .

Phase Shift:

$\frac{4 (0)}{4 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

$4$ out of

$4 π$ .

Phase Shift:

$\frac{4 (0)}{4 (π)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift:

$\frac{4 \cdot 0}{4 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift:

$\frac{0}{π}$

Phase Shift:

$\frac{0}{π}$

Phase Shift:

$\frac{0}{π}$

Step 4.4

Divide

$0$ by

$π$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$\frac{1}{2}$

Phase Shift: None

Vertical Shift: None

Step 6