Graph y=5sin(1/2x) - MathMaster

$y = 5 \sin (\frac{1}{2} x)$

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

$a = 5$

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

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $5$

Find the period using the formula $\frac{2 π}{| b |}$ .

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

Period: $\frac{2 π}{| b |}$

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

Period: $\frac{2 π}{| \frac{1}{2} |}$

Solve the equation.

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

Period: $\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Period: $2 π \cdot 2$

Multiply $2$ by $2$ .

Period: $4 π$

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: $0$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $4 π$

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

Vertical Shift: $0$

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.

$f (0) = 5 \sin (\frac{0}{2})$

Simplify the result.

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Divide $0$ by $2$ .

$f (0) = 5 \sin (0)$

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

$f (0) = 5 \cdot 0$

Multiply $5$ by $0$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π$ .

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

$f (π) = 5 \sin (\frac{π}{2})$

Simplify the result.

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

$f (π) = 5 \cdot 1$

Multiply $5$ by $1$ .

$f (π) = 5$

The final answer is $5$ .

$5$

Find the point at $x = 2 π$ .

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

$f (2 π) = 5 \sin (\frac{2 π}{2})$

Simplify the result.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$f (2 π) = 5 \sin (\frac{2 π}{2})$

Divide $π$ by $1$ .

$f (2 π) = 5 \sin (π)$

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

$f (2 π) = 5 \sin (0)$

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

$f (2 π) = 5 \cdot 0$

Multiply $5$ by $0$ .

$f (2 π) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 3 π$ .

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

$f (3 π) = 5 \sin (\frac{3 π}{2})$

Simplify the result.

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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.

$f (3 π) = 5 (- \sin (\frac{π}{2}))$

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

$f (3 π) = 5 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (3 π) = 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$f (3 π) = - 5$

The final answer is $- 5$ .

$- 5$

Find the point at $x = 4 π$ .

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

$f (4 π) = 5 \sin (\frac{4 π}{2})$

Simplify the result.

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

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Factor $2$ out of $4 π$ .

$f (4 π) = 5 \sin (\frac{2 (2 π)}{2})$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (4 π) = 5 \sin (\frac{2 (2 π)}{2 (1)})$

Cancel the common factor.

$f (4 π) = 5 \sin (\frac{2 (2 π)}{2 \cdot 1})$

Rewrite the expression.

$f (4 π) = 5 \sin (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (4 π) = 5 \sin (2 π)$

$2 π$ is a full rotation so replace with $0$ .

$f (4 π) = 5 \sin (0)$

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

$f (4 π) = 5 \cdot 0$

Multiply $5$ by $0$ .

$f (4 π) = 0$

The final answer is $0$ .

$0$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ π & 5 \\ 2 π & 0 \\ 3 π & - 5 \\ 4 π & 0 \end{array}$

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

Amplitude: $5$

Period: $4 π$

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

Vertical Shift: $0$

$\begin{array}{c} x & f (x) \\ 0 & 0 \\ π & 5 \\ 2 π & 0 \\ 3 π & - 5 \\ 4 π & 0 \end{array}$

Graph y=5sin(1/2x)

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