Pre-Algebra Examples

y = - 13 sin (\frac{π}{12} x - 1) + 50

$y = - 13 \sin (\frac{π}{12} x - 1) + 50$

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 = - 13

$a = - 13$

b = \frac{π}{12}

$b = \frac{π}{12}$

c = 1

$c = 1$

d = 50

$d = 50$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

13

$13$

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

- 13 sin (\frac{π x}{12} - 1)

$- 13 \sin (\frac{π x}{12} - 1)$ .

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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

\frac{π}{12}

$\frac{π}{12}$ in the formula for period.

\frac{2 π}{∣ ∣ \frac{π}{12} ∣ ∣}

$\frac{2 π}{| \frac{π}{12} |}$

Step 3.1.3

\frac{π}{12}

$\frac{π}{12}$ is approximately

0.26179938

$0.26179938$ which is positive so remove the absolute value

\frac{2 π}{\frac{π}{12}}

$\frac{2 π}{\frac{π}{12}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

2 π \frac{12}{π}

$2 π \frac{12}{π}$

Step 3.1.5

Cancel the common factor of

π

$π$ .

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Step 3.1.5.1

Factor

π

$π$ out of

2 π

$2 π$ .

π \cdot 2 \frac{12}{π}

$π \cdot 2 \frac{12}{π}$

Step 3.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{12}{π}$

Step 3.1.5.3

Rewrite the expression.

$2 \cdot 12$

Step 3.1.6

Multiply

$2$ by

$12$ .

$24$

Step 3.2

Find the period of

$50$ .

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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

$\frac{π}{12}$ in the formula for period.

$\frac{2 π}{| \frac{π}{12} |}$

Step 3.2.3

$\frac{π}{12}$ is approximately

$0.26179938$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{12}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{12}{π}$

Step 3.2.5

Cancel the common factor of

$π$ .

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Step 3.2.5.1

Factor

$π$ out of

$2 π$ .

$π \cdot 2 \frac{12}{π}$

Step 3.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{12}{π}$

Step 3.2.5.3

Rewrite the expression.

$2 \cdot 12$

Step 3.2.6

Multiply

$2$ by

$12$ .

$24$

Step 3.3

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

$24$

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:

$\frac{1}{\frac{π}{12}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$1 (\frac{12}{π})$

Step 4.4

Multiply

$\frac{12}{π}$ by

$1$ .

Phase Shift:

$\frac{12}{π}$

Phase Shift:

$\frac{12}{π}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$13$

Period:

$24$

Phase Shift:

$\frac{12}{π}$ (

$\frac{12}{π}$ to the right)

Vertical Shift:

$50$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at

$x = 6 + \frac{12}{π}$ .

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Step 6.1.1

Replace the variable

$x$ with

$6 + \frac{12}{π}$ in the expression.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π (6 + \frac{12}{π})}{12} - 1) + 50$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Simplify each term.

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Step 6.1.2.1.1.1

Cancel the common factor of

$6 + \frac{12}{π}$ and

$12$ .

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Step 6.1.2.1.1.1.1

Factor

$6$ out of

$π (6 + \frac{12}{π})$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{6 (π (1 + \frac{2}{π}))}{12} - 1) + 50$

Step 6.1.2.1.1.1.2

Cancel the common factors.

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Step 6.1.2.1.1.1.2.1

Factor

$6$ out of

$12$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{6 (π (1 + \frac{2}{π}))}{6 (2)} - 1) + 50$

Step 6.1.2.1.1.1.2.2

Cancel the common factor.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{6 (π (1 + \frac{2}{π}))}{6 \cdot 2} - 1) + 50$

Step 6.1.2.1.1.1.2.3

Rewrite the expression.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π (1 + \frac{2}{π})}{2} - 1) + 50$

Step 6.1.2.1.1.2

Simplify the numerator.

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Step 6.1.2.1.1.2.1

Write

$1$ as a fraction with a common denominator.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π (\frac{π}{π} + \frac{2}{π})}{2} - 1) + 50$

Step 6.1.2.1.1.2.2

Combine the numerators over the common denominator.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π (\frac{π + 2}{π})}{2} - 1) + 50$

Step 6.1.2.1.1.3

Combine

$π$ and

$\frac{π + 2}{π}$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{\frac{π (π + 2)}{π}}{2} - 1) + 50$

Step 6.1.2.1.1.4

Reduce the expression by cancelling the common factors.

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Step 6.1.2.1.1.4.1

Reduce the expression

$\frac{π (π + 2)}{π}$ by cancelling the common factors.

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Step 6.1.2.1.1.4.1.1

Cancel the common factor.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{\frac{π (π + 2)}{π}}{2} - 1) + 50$

Step 6.1.2.1.1.4.1.2

Rewrite the expression.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{\frac{π + 2}{1}}{2} - 1) + 50$

Step 6.1.2.1.1.4.2

Divide

$π + 2$ by

$1$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 2}{2} - 1) + 50$

Step 6.1.2.1.2

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 2}{2} - 1 \cdot \frac{2}{2}) + 50$

Step 6.1.2.1.3

Combine

$- 1$ and

$\frac{2}{2}$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 2}{2} + \frac{- 1 \cdot 2}{2}) + 50$

Step 6.1.2.1.4

Combine the numerators over the common denominator.

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 2 - 1 \cdot 2}{2}) + 50$

Step 6.1.2.1.5

Simplify the numerator.

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Step 6.1.2.1.5.1

Multiply

$- 1$ by

$2$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 2 - 2}{2}) + 50$

Step 6.1.2.1.5.2

Subtract

$2$ from

$2$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π + 0}{2}) + 50$

Step 6.1.2.1.5.3

Add

$π$ and

$0$ .

$f (6 + \frac{12}{π}) = - 13 \sin (\frac{π}{2}) + 50$

Step 6.1.2.1.6

The exact value of

$\sin (\frac{π}{2})$ is

$1$ .

$f (6 + \frac{12}{π}) = - 13 \cdot 1 + 50$

Step 6.1.2.1.7

Multiply

$- 13$ by

$1$ .

$f (6 + \frac{12}{π}) = - 13 + 50$

Step 6.1.2.2

Add

$- 13$ and

$50$ .

$f (6 + \frac{12}{π}) = 37$

Step 6.1.2.3

The final answer is

$37$ .

$37$

Step 6.2

List the points in a table.

$\begin{array}{c} x & f (x) \\ 6 + \frac{12}{π} & 37 \\ 30 + \frac{12}{π} & 37 \\ 54 + \frac{12}{π} & 37 \\ 78 + \frac{12}{π} & 37 \\ 102 + \frac{12}{π} & 37 \end{array}$

Step 7

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

Amplitude:

$13$

Period:

$24$

Phase Shift:

$\frac{12}{π}$ (

$\frac{12}{π}$ to the right)

Vertical Shift:

$50$

$\begin{array}{c} x & f (x) \\ 6 + \frac{12}{π} & 37 \\ 30 + \frac{12}{π} & 37 \\ 54 + \frac{12}{π} & 37 \\ 78 + \frac{12}{π} & 37 \\ 102 + \frac{12}{π} & 37 \end{array}$

Step 8