Trigonometry Examples

$y = 2 \sin (- 4 x)$

Step 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.

$a = 2$

$b = - 4$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

Find the period of $2 \sin (- 4 x)$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $- 4$ in the formula for period.

$\frac{2 π}{| - 4 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$\frac{2 π}{4}$

Step 3.4

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

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{0}{- 4}$

Step 4.3

Divide $0$ by $- 4$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{π}{2}$

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 $x = 0$ .

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Step 6.1.1

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

$f (0) = 2 \sin (- 4 \cdot 0)$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Multiply $- 4$ by $0$ .

$f (0) = 2 \sin (0)$

Step 6.1.2.2

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

$f (0) = 2 \cdot 0$

Step 6.1.2.3

Multiply $2$ by $0$ .

$f (0) = 0$

Step 6.1.2.4

The final answer is $0$ .

$0$

Step 6.2

Find the point at $x = \frac{π}{8}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{π}{8}$ in the expression.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1

Factor $4$ out of $- 4$ .

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

Step 6.2.2.1.2

Factor $4$ out of $8$ .

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

Step 6.2.2.1.3

Cancel the common factor.

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

Step 6.2.2.1.4

Rewrite the expression.

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

Step 6.2.2.2

Rewrite $- 1 \frac{π}{2}$ as $- \frac{π}{2}$ .

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

Step 6.2.2.3

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

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

Step 6.2.2.4

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.

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

Step 6.2.2.5

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

$f (\frac{π}{8}) = 2 (- 1 \cdot 1)$

Step 6.2.2.6

Multiply $2 (- 1 \cdot 1)$ .

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Step 6.2.2.6.1

Multiply $- 1$ by $1$ .

$f (\frac{π}{8}) = 2 \cdot - 1$

Step 6.2.2.6.2

Multiply $2$ by $- 1$ .

$f (\frac{π}{8}) = - 2$

Step 6.2.2.7

The final answer is $- 2$ .

$- 2$

Step 6.3

Find the point at $x = \frac{π}{4}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{π}{4}$ in the expression.

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Cancel the common factor of $4$ .

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Step 6.3.2.1.1

Factor $4$ out of $- 4$ .

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

Step 6.3.2.1.2

Cancel the common factor.

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

Step 6.3.2.1.3

Rewrite the expression.

$f (\frac{π}{4}) = 2 \sin (- 1 π)$

Step 6.3.2.2

Rewrite $- 1 π$ as $- π$ .

$f (\frac{π}{4}) = 2 \sin (- π)$

Step 6.3.2.3

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

$f (\frac{π}{4}) = 2 \sin (0)$

Step 6.3.2.4

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

$f (\frac{π}{4}) = 2 \cdot 0$

Step 6.3.2.5

Multiply $2$ by $0$ .

$f (\frac{π}{4}) = 0$

Step 6.3.2.6

The final answer is $0$ .

$0$

Step 6.4

Find the point at $x = \frac{3 π}{8}$ .

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Step 6.4.1

Replace the variable $x$ with $\frac{3 π}{8}$ in the expression.

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $4$ .

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Step 6.4.2.1.1

Factor $4$ out of $- 4$ .

$f (\frac{3 π}{8}) = 2 \sin (4 (- 1) (\frac{3 π}{8}))$

Step 6.4.2.1.2

Factor $4$ out of $8$ .

$f (\frac{3 π}{8}) = 2 \sin (4 \cdot (- 1 \frac{3 π}{4 \cdot 2}))$

Step 6.4.2.1.3

Cancel the common factor.

$f (\frac{3 π}{8}) = 2 \sin (4 \cdot (- 1 \frac{3 π}{4 \cdot 2}))$

Step 6.4.2.1.4

Rewrite the expression.

$f (\frac{3 π}{8}) = 2 \sin (- 1 \frac{3 π}{2})$

Step 6.4.2.2

Rewrite $- 1 \frac{3 π}{2}$ as $- \frac{3 π}{2}$ .

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

Step 6.4.2.3

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

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

Step 6.4.2.4

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

$f (\frac{3 π}{8}) = 2 \cdot 1$

Step 6.4.2.5

Multiply $2$ by $1$ .

$f (\frac{3 π}{8}) = 2$

Step 6.4.2.6

The final answer is $2$ .

$2$

Step 6.5

Find the point at $x = \frac{π}{2}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{π}{2}$ in the expression.

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Cancel the common factor of $2$ .

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Step 6.5.2.1.1

Factor $2$ out of $- 4$ .

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

Step 6.5.2.1.2

Cancel the common factor.

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

Step 6.5.2.1.3

Rewrite the expression.

$f (\frac{π}{2}) = 2 \sin (- 2 π)$

Step 6.5.2.2

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

$f (\frac{π}{2}) = 2 \sin (0)$

Step 6.5.2.3

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

$f (\frac{π}{2}) = 2 \cdot 0$

Step 6.5.2.4

Multiply $2$ by $0$ .

$f (\frac{π}{2}) = 0$

Step 6.5.2.5

The final answer is $0$ .

$0$

Step 6.6

List the points in a table.

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

Step 7

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

Amplitude: $2$

Period: $\frac{π}{2}$

Phase Shift: None

Vertical Shift: None

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

Step 8