Precalculus Examples

f (θ) = sin (4 θ)

$f (θ) = \sin (4 θ)$

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

$b = 4$

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 x)

$\sin (4 x)$ .

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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

$4$ in the formula for period.

\frac{2 π}{| 4 |}

$\frac{2 π}{| 4 |}$

Step 3.3

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

0

$0$ and

4

$4$ is

4

$4$ .

\frac{2 π}{4}

$\frac{2 π}{4}$

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}

$\frac{2 (π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

2

$2$ out of

4

$4$ .

\frac{2 π}{2 \cdot 2}

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

\frac{2 π}{2 \cdot 2}

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

\frac{π}{2}

$\frac{π}{2}$

\frac{π}{2}

$\frac{π}{2}$

\frac{π}{2}

$\frac{π}{2}$

\frac{π}{2}

$\frac{π}{2}$

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}{4}

$\frac{0}{4}$

Step 4.3

Divide

0

$0$ by

4

$4$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

1

$1$

Period:

\frac{π}{2}

$\frac{π}{2}$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at

x = 0

$x = 0$ .

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Step 6.1.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = sin (4 (0))

$f (0) = \sin (4 (0))$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply

4

$4$ by

0

$0$ .

f (0) = sin (0)

$f (0) = \sin (0)$

Step 6.1.2.2

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

f (0) = 0

$f (0) = 0$

Step 6.1.2.3

The final answer is

0

$0$ .

0

$0$

0

$0$

0

$0$

Step 6.2

Find the point at

x = \frac{π}{8}

$x = \frac{π}{8}$ .

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Step 6.2.1

Replace the variable

x

$x$ with

\frac{π}{8}

$\frac{π}{8}$ in the expression.

f (\frac{π}{8}) = sin (4 (\frac{π}{8}))

$f (\frac{π}{8}) = \sin (4 (\frac{π}{8}))$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Cancel the common factor of

4

$4$ .

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Step 6.2.2.1.1

Factor

4

$4$ out of

8

$8$ .

f (\frac{π}{8}) = sin (4 (\frac{π}{4 (2)}))

$f (\frac{π}{8}) = \sin (4 (\frac{π}{4 (2)}))$

Step 6.2.2.1.2

Cancel the common factor.

f (\frac{π}{8}) = sin (4 (\frac{π}{4 \cdot 2}))

$f (\frac{π}{8}) = \sin (4 (\frac{π}{4 \cdot 2}))$

Step 6.2.2.1.3

Rewrite the expression.

f (\frac{π}{8}) = sin (\frac{π}{2})

$f (\frac{π}{8}) = \sin (\frac{π}{2})$

f (\frac{π}{8}) = sin (\frac{π}{2})

$f (\frac{π}{8}) = \sin (\frac{π}{2})$

Step 6.2.2.2

The exact value of

sin (\frac{π}{2})

$\sin (\frac{π}{2})$ is

1

$1$ .

f (\frac{π}{8}) = 1

$f (\frac{π}{8}) = 1$

Step 6.2.2.3

The final answer is

1

$1$ .

1

$1$

1

$1$

1

$1$

Step 6.3

Find the point at

x = \frac{π}{4}

$x = \frac{π}{4}$ .

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Step 6.3.1

Replace the variable

x

$x$ with

\frac{π}{4}

$\frac{π}{4}$ in the expression.

f (\frac{π}{4}) = sin (4 (\frac{π}{4}))

$f (\frac{π}{4}) = \sin (4 (\frac{π}{4}))$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of

4

$4$ .

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Step 6.3.2.1.1

Cancel the common factor.

f (\frac{π}{4}) = sin (4 (\frac{π}{4}))

$f (\frac{π}{4}) = \sin (4 (\frac{π}{4}))$

Step 6.3.2.1.2

Rewrite the expression.

f (\frac{π}{4}) = sin (π)

$f (\frac{π}{4}) = \sin (π)$

f (\frac{π}{4}) = sin (π)

$f (\frac{π}{4}) = \sin (π)$

Step 6.3.2.2

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

f (\frac{π}{4}) = sin (0)

$f (\frac{π}{4}) = \sin (0)$

Step 6.3.2.3

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

f (\frac{π}{4}) = 0

$f (\frac{π}{4}) = 0$

Step 6.3.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

0

$0$

Step 6.4

Find the point at

x = \frac{3 π}{8}

$x = \frac{3 π}{8}$ .

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Step 6.4.1

Replace the variable

x

$x$ with

\frac{3 π}{8}

$\frac{3 π}{8}$ in the expression.

f (\frac{3 π}{8}) = sin (4 (\frac{3 π}{8}))

$f (\frac{3 π}{8}) = \sin (4 (\frac{3 π}{8}))$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of

4

$4$ .

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Step 6.4.2.1.1

Factor

4

$4$ out of

8

$8$ .

f (\frac{3 π}{8}) = sin (4 (\frac{3 π}{4 (2)}))

$f (\frac{3 π}{8}) = \sin (4 (\frac{3 π}{4 (2)}))$

Step 6.4.2.1.2

Cancel the common factor.

f (\frac{3 π}{8}) = sin (4 (\frac{3 π}{4 \cdot 2}))

$f (\frac{3 π}{8}) = \sin (4 (\frac{3 π}{4 \cdot 2}))$

Step 6.4.2.1.3

Rewrite the expression.

f (\frac{3 π}{8}) = sin (\frac{3 π}{2})

$f (\frac{3 π}{8}) = \sin (\frac{3 π}{2})$

f (\frac{3 π}{8}) = sin (\frac{3 π}{2})

$f (\frac{3 π}{8}) = \sin (\frac{3 π}{2})$

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.

f (\frac{3 π}{8}) = - sin (\frac{π}{2})

$f (\frac{3 π}{8}) = - \sin (\frac{π}{2})$

Step 6.4.2.3

The exact value of

sin (\frac{π}{2})

$\sin (\frac{π}{2})$ is

1

$1$ .

f (\frac{3 π}{8}) = - 1 \cdot 1

$f (\frac{3 π}{8}) = - 1 \cdot 1$

Step 6.4.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

f (\frac{3 π}{8}) = - 1

$f (\frac{3 π}{8}) = - 1$

Step 6.4.2.5

The final answer is

- 1

$- 1$ .

- 1

$- 1$

- 1

$- 1$

- 1

$- 1$

Step 6.5

Find the point at

x = \frac{π}{2}

$x = \frac{π}{2}$ .

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Step 6.5.1

Replace the variable

x

$x$ with

\frac{π}{2}

$\frac{π}{2}$ in the expression.

f (\frac{π}{2}) = sin (4 (\frac{π}{2}))

$f (\frac{π}{2}) = \sin (4 (\frac{π}{2}))$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of

2

$2$ .

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Step 6.5.2.1.1

Factor

2

$2$ out of

4

$4$ .

f (\frac{π}{2}) = sin (2 (2) (\frac{π}{2}))

$f (\frac{π}{2}) = \sin (2 (2) (\frac{π}{2}))$

Step 6.5.2.1.2

Cancel the common factor.

f (\frac{π}{2}) = sin (2 \cdot (2 (\frac{π}{2})))

$f (\frac{π}{2}) = \sin (2 \cdot (2 (\frac{π}{2})))$

Step 6.5.2.1.3

Rewrite the expression.

f (\frac{π}{2}) = sin (2 π)

$f (\frac{π}{2}) = \sin (2 π)$

f (\frac{π}{2}) = sin (2 π)

$f (\frac{π}{2}) = \sin (2 π)$

Step 6.5.2.2

Subtract full rotations of

2 π

$2 π$ until the angle is greater than or equal to

0

$0$ and less than

2 π

$2 π$ .

f (\frac{π}{2}) = sin (0)

$f (\frac{π}{2}) = \sin (0)$

Step 6.5.2.3

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

f (\frac{π}{2}) = 0

$f (\frac{π}{2}) = 0$

Step 6.5.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

0

$0$

Step 6.6

List the points in a table.

\begin{matrix} x & f (x) 0 & 0 \frac{π}{8} & 1 \frac{π}{4} & 0 \frac{3 π}{8} & - 1 \frac{π}{2} & 0 \end{matrix}

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{8} & 1 \\ \frac{π}{4} & 0 \\ \frac{3 π}{8} & - 1 \\ \frac{π}{2} & 0 \end{array}$

\begin{matrix} x & f (x) 0 & 0 \frac{π}{8} & 1 \frac{π}{4} & 0 \frac{3 π}{8} & - 1 \frac{π}{2} & 0 \end{matrix}

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{8} & 1 \\ \frac{π}{4} & 0 \\ \frac{3 π}{8} & - 1 \\ \frac{π}{2} & 0 \end{array}$

Step 7

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

Amplitude:

1

$1$

Period:

\frac{π}{2}

$\frac{π}{2}$

Phase Shift: None

Vertical Shift: None

\begin{matrix} x & f (x) 0 & 0 \frac{π}{8} & 1 \frac{π}{4} & 0 \frac{3 π}{8} & - 1 \frac{π}{2} & 0 \end{matrix}

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ \frac{π}{8} & 1 \\ \frac{π}{4} & 0 \\ \frac{3 π}{8} & - 1 \\ \frac{π}{2} & 0 \end{array}$

Step 8

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