Graph y=-3sin(1/2x)-1

$y = - 3 \sin (\frac{1}{2} x) - 1$

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

$a = - 3$

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

$c = 0$

$d = - 1$

Find the amplitude $| a |$ .

Amplitude: $3$

Find the period of $- 3 \sin (\frac{x}{2})$ .

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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: $\frac{0}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 2$

Multiply $0$ by $2$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $- 1$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $4 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $- 1$

Select a few points to graph.

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Find the point at $x = 0$ .

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Replace the variable $x$ with $0$ in the expression.

$f (0) = - 3 \sin (\frac{0}{2}) - 1$

Simplify the result.

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Simplify each term.

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Divide $0$ by $2$ .

$f (0) = - 3 \sin (0) - 1$

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

$f (0) = - 3 \cdot 0 - 1$

Multiply $- 3$ by $0$ .

$f (0) = 0 - 1$

Subtract $1$ from $0$ .

$f (0) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = π$ .

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

$f (π) = - 3 \sin (\frac{π}{2}) - 1$

Simplify the result.

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Simplify each term.

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

$f (π) = - 3 \cdot 1 - 1$

Multiply $- 3$ by $1$ .

$f (π) = - 3 - 1$

Subtract $1$ from $- 3$ .

$f (π) = - 4$

The final answer is $- 4$ .

$- 4$

Find the point at $x = 2 π$ .

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

$f (2 π) = - 3 \sin (\frac{2 π}{2}) - 1$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$f (2 π) = - 3 \sin (\frac{2 π}{2}) - 1$

Divide $π$ by $1$ .

$f (2 π) = - 3 \sin (π) - 1$

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

$f (2 π) = - 3 \sin (0) - 1$

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

$f (2 π) = - 3 \cdot 0 - 1$

Multiply $- 3$ by $0$ .

$f (2 π) = 0 - 1$

Subtract $1$ from $0$ .

$f (2 π) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = 3 π$ .

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

$f (3 π) = - 3 \sin (\frac{3 π}{2}) - 1$

Simplify the result.

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Simplify each term.

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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.

$f (3 π) = - 3 (- \sin (\frac{π}{2})) - 1$

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

$f (3 π) = - 3 (- 1 \cdot 1) - 1$

Multiply $- 1$ by $1$ .

$f (3 π) = - 3 \cdot - 1 - 1$

Multiply $- 3$ by $- 1$ .

$f (3 π) = 3 - 1$

Subtract $1$ from $3$ .

$f (3 π) = 2$

The final answer is $2$ .

$2$

Find the point at $x = 4 π$ .

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

$f (4 π) = - 3 \sin (\frac{4 π}{2}) - 1$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4 π$ .

$f (4 π) = - 3 \sin (\frac{2 (2 π)}{2}) - 1$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (4 π) = - 3 \sin (\frac{2 (2 π)}{2 (1)}) - 1$

Cancel the common factor.

$f (4 π) = - 3 \sin (\frac{2 (2 π)}{2 \cdot 1}) - 1$

Rewrite the expression.

$f (4 π) = - 3 \sin (\frac{2 π}{1}) - 1$

Divide $2 π$ by $1$ .

$f (4 π) = - 3 \sin (2 π) - 1$

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

$f (4 π) = - 3 \sin (0) - 1$

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

$f (4 π) = - 3 \cdot 0 - 1$

Multiply $- 3$ by $0$ .

$f (4 π) = 0 - 1$

Subtract $1$ from $0$ .

$f (4 π) = - 1$

The final answer is $- 1$ .

$- 1$

List the points in a table.

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

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

Amplitude: $3$

Period: $4 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $- 1$

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

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