Graph y=6sin(1/2x-8)+0

$y = 6 \sin (\frac{1}{2} x - 8) + 0$

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

$a = 6$

$b = \frac{1}{2}$

$c = 8$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $6$

Find the period of $6 \sin (\frac{x}{2} - 8)$ .

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

Find the phase shift using the formula $\frac{c}{b}$ .

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The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{8}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $8 \cdot 2$

Multiply $8$ by $2$ .

Phase Shift: $16$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $6$

Period: $4 π$

Phase Shift: $16$ ( $16$ to the right)

Vertical Shift: $0$

Select a few points to graph.

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Find the point at $x = 16$ .

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Replace the variable $x$ with $16$ in the expression.

$f (16) = 6 \sin (\frac{16}{2} - 8)$

Simplify the result.

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Divide $16$ by $2$ .

$f (16) = 6 \sin (8 - 8)$

Subtract $8$ from $8$ .

$f (16) = 6 \sin (0)$

The exact value of $\sin (0)$ is $0$ .

$f (16) = 6 \cdot 0$

Multiply $6$ by $0$ .

$f (16) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π + 16$ .

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Replace the variable $x$ with $π + 16$ in the expression.

$f (π + 16) = 6 \sin (\frac{π + 16}{2} - 8)$

Simplify the result.

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To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (π + 16) = 6 \sin (\frac{π + 16}{2} - 8 \cdot \frac{2}{2})$

Combine fractions.

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Combine $- 8$ and $\frac{2}{2}$ .

$f (π + 16) = 6 \sin (\frac{π + 16}{2} + \frac{- 8 \cdot 2}{2})$

Combine the numerators over the common denominator.

$f (π + 16) = 6 \sin (\frac{π + 16 - 8 \cdot 2}{2})$

Simplify the numerator.

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Multiply $- 8$ by $2$ .

$f (π + 16) = 6 \sin (\frac{π + 16 - 16}{2})$

Subtract $16$ from $16$ .

$f (π + 16) = 6 \sin (\frac{π + 0}{2})$

Add $π$ and $0$ .

$f (π + 16) = 6 \sin (\frac{π}{2})$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (π + 16) = 6 \cdot 1$

Multiply $6$ by $1$ .

$f (π + 16) = 6$

The final answer is $6$ .

$6$

Find the point at $x = 2 π + 16$ .

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Replace the variable $x$ with $2 π + 16$ in the expression.

$f (2 π + 16) = 6 \sin (\frac{2 π + 16}{2} - 8)$

Simplify the result.

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Cancel the common factor of $2 π + 16$ and $2$ .

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Factor $2$ out of $2 π$ .

$f (2 π + 16) = 6 \sin (\frac{2 (π) + 16}{2} - 8)$

Factor $2$ out of $16$ .

$f (2 π + 16) = 6 \sin (\frac{2 (π) + 2 \cdot 8}{2} - 8)$

Factor $2$ out of $2 (π) + 2 (8)$ .

$f (2 π + 16) = 6 \sin (\frac{2 (π + 8)}{2} - 8)$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (2 π + 16) = 6 \sin (\frac{2 (π + 8)}{2 (1)} - 8)$

Cancel the common factor.

$f (2 π + 16) = 6 \sin (\frac{2 (π + 8)}{2 \cdot 1} - 8)$

Rewrite the expression.

$f (2 π + 16) = 6 \sin (\frac{π + 8}{1} - 8)$

Divide $π + 8$ by $1$ .

$f (2 π + 16) = 6 \sin (π + 8 - 8)$

Simplify by subtracting numbers.

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Subtract $8$ from $8$ .

$f (2 π + 16) = 6 \sin (π + 0)$

Add $π$ and $0$ .

$f (2 π + 16) = 6 \sin (π)$

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

$f (2 π + 16) = 6 \sin (0)$

The exact value of $\sin (0)$ is $0$ .

$f (2 π + 16) = 6 \cdot 0$

Multiply $6$ by $0$ .

$f (2 π + 16) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 3 π + 16$ .

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Replace the variable $x$ with $3 π + 16$ in the expression.

$f (3 π + 16) = 6 \sin (\frac{3 π + 16}{2} - 8)$

Simplify the result.

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To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (3 π + 16) = 6 \sin (\frac{3 π + 16}{2} - 8 \cdot \frac{2}{2})$

Combine fractions.

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Combine $- 8$ and $\frac{2}{2}$ .

$f (3 π + 16) = 6 \sin (\frac{3 π + 16}{2} + \frac{- 8 \cdot 2}{2})$

Combine the numerators over the common denominator.

$f (3 π + 16) = 6 \sin (\frac{3 π + 16 - 8 \cdot 2}{2})$

Simplify the numerator.

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Multiply $- 8$ by $2$ .

$f (3 π + 16) = 6 \sin (\frac{3 π + 16 - 16}{2})$

Subtract $16$ from $16$ .

$f (3 π + 16) = 6 \sin (\frac{3 π + 0}{2})$

Add $3 π$ and $0$ .

$f (3 π + 16) = 6 \sin (\frac{3 π}{2})$

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.

$f (3 π + 16) = 6 (- \sin (\frac{π}{2}))$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (3 π + 16) = 6 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (3 π + 16) = 6 \cdot - 1$

Multiply $6$ by $- 1$ .

$f (3 π + 16) = - 6$

The final answer is $- 6$ .

$- 6$

Find the point at $x = 4 π + 16$ .

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Replace the variable $x$ with $4 π + 16$ in the expression.

$f (4 π + 16) = 6 \sin (\frac{4 π + 16}{2} - 8)$

Simplify the result.

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Cancel the common factor of $4 π + 16$ and $2$ .

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Factor $2$ out of $4 π$ .

$f (4 π + 16) = 6 \sin (\frac{2 (2 π) + 16}{2} - 8)$

Factor $2$ out of $16$ .

$f (4 π + 16) = 6 \sin (\frac{2 (2 π) + 2 \cdot 8}{2} - 8)$

Factor $2$ out of $2 (2 π) + 2 (8)$ .

$f (4 π + 16) = 6 \sin (\frac{2 (2 π + 8)}{2} - 8)$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (4 π + 16) = 6 \sin (\frac{2 (2 π + 8)}{2 (1)} - 8)$

Cancel the common factor.

$f (4 π + 16) = 6 \sin (\frac{2 (2 π + 8)}{2 \cdot 1} - 8)$

Rewrite the expression.

$f (4 π + 16) = 6 \sin (\frac{2 π + 8}{1} - 8)$

Divide $2 π + 8$ by $1$ .

$f (4 π + 16) = 6 \sin (2 π + 8 - 8)$

Simplify by subtracting numbers.

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Subtract $8$ from $8$ .

$f (4 π + 16) = 6 \sin (2 π + 0)$

Add $2 π$ and $0$ .

$f (4 π + 16) = 6 \sin (2 π)$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (4 π + 16) = 6 \sin (0)$

The exact value of $\sin (0)$ is $0$ .

$f (4 π + 16) = 6 \cdot 0$

Multiply $6$ by $0$ .

$f (4 π + 16) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 16 & 0 \\ π + 16 & 6 \\ 2 π + 16 & 0 \\ 3 π + 16 & - 6 \\ 4 π + 16 & 0 \end{array}$

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

Amplitude: $6$

Period: $4 π$

Phase Shift: $16$ ( $16$ to the right)

Vertical Shift: $0$

$\begin{array}{c} x & f (x) \\ 16 & 0 \\ π + 16 & 6 \\ 2 π + 16 & 0 \\ 3 π + 16 & - 6 \\ 4 π + 16 & 0 \end{array}$

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