Trigonometry Examples

$y = - \frac{1}{2} \cdot \sin (4 (x + \frac{π}{4})) + 1$

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.

$a = - \frac{1}{2}$

$b = 4$

$c = - π$

$d = 1$

Step 2

Find the amplitude $| a |$ .

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

Step 3

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

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Step 3.1

Find the period of $- \frac{\sin (4 x + π)}{2}$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

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

$\frac{2 π}{| 4 |}$

Step 3.1.3

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

$\frac{2 π}{4}$

Step 3.1.4

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

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Step 3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.2

Find the period of $1$ .

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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 $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 3.2.3

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

$\frac{2 π}{4}$

Step 3.2.4

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

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Step 3.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 3.3

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

$\frac{π}{2}$

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: $\frac{- π}{4}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{4}$

Step 5

List the properties of the trigonometric function.

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

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: $1$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - \frac{π}{4}$ .

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Step 6.1.1

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

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

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 the numerator.

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Step 6.1.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.1.2.1.1.1.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

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

Step 6.1.2.1.1.1.2

Cancel the common factor.

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

Step 6.1.2.1.1.1.3

Rewrite the expression.

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

Step 6.1.2.1.1.2

Add $- π$ and $π$ .

$f (- \frac{π}{4}) = - \frac{\sin (0)}{2} + 1$

Step 6.1.2.1.1.3

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

$f (- \frac{π}{4}) = - \frac{0}{2} + 1$

Step 6.1.2.1.2

Divide $0$ by $2$ .

$f (- \frac{π}{4}) = - 0 + 1$

Step 6.1.2.1.3

Multiply $- 1$ by $0$ .

$f (- \frac{π}{4}) = 0 + 1$

Step 6.1.2.2

Add $0$ and $1$ .

$f (- \frac{π}{4}) = 1$

Step 6.1.2.3

The final answer is $1$ .

$1$

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.

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

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify the numerator.

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Step 6.2.2.1.1

Simplify each term.

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Step 6.2.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1.1.1

Move the leading negative in $- \frac{π}{8}$ into the numerator.

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

Step 6.2.2.1.1.1.2

Factor $4$ out of $8$ .

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

Step 6.2.2.1.1.1.3

Cancel the common factor.

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

Step 6.2.2.1.1.1.4

Rewrite the expression.

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

Step 6.2.2.1.1.2

Move the negative in front of the fraction.

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

Step 6.2.2.1.2

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

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

Step 6.2.2.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.2.1.4

Combine the numerators over the common denominator.

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

Step 6.2.2.1.5

Simplify the numerator.

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Step 6.2.2.1.5.1

Move $2$ to the left of $π$ .

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

Step 6.2.2.1.5.2

Add $- π$ and $2 π$ .

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

Step 6.2.2.1.6

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

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

Step 6.2.2.2

Simplify the expression.

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Step 6.2.2.2.1

Write $1$ as a fraction with a common denominator.

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

Step 6.2.2.2.2

Combine the numerators over the common denominator.

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

Step 6.2.2.2.3

Add $- 1$ and $2$ .

$f (- \frac{π}{8}) = \frac{1}{2}$

Step 6.2.2.3

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 6.3

Find the point at $x = 0$ .

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Step 6.3.1

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

$f (0) = - \frac{\sin (4 (0) + π)}{2} + 1$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Simplify the numerator.

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Step 6.3.2.1.1.1

Multiply $4$ by $0$ .

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

Step 6.3.2.1.1.2

Add $0$ and $π$ .

$f (0) = - \frac{\sin (π)}{2} + 1$

Step 6.3.2.1.1.3

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

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

Step 6.3.2.1.1.4

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

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

Step 6.3.2.1.2

Divide $0$ by $2$ .

$f (0) = - 0 + 1$

Step 6.3.2.1.3

Multiply $- 1$ by $0$ .

$f (0) = 0 + 1$

Step 6.3.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 6.3.2.3

The final answer is $1$ .

$1$

Step 6.4

Find the point at $x = \frac{π}{8}$ .

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Step 6.4.1

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

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Simplify the numerator.

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Step 6.4.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.4.2.1.1.1.1

Factor $4$ out of $8$ .

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

Step 6.4.2.1.1.1.2

Cancel the common factor.

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

Step 6.4.2.1.1.1.3

Rewrite the expression.

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

Step 6.4.2.1.1.2

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

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

Step 6.4.2.1.1.3

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.4.2.1.1.4

Combine the numerators over the common denominator.

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

Step 6.4.2.1.1.5

Simplify the numerator.

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Step 6.4.2.1.1.5.1

Move $2$ to the left of $π$ .

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

Step 6.4.2.1.1.5.2

Add $π$ and $2 π$ .

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

Step 6.4.2.1.1.6

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.

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

Step 6.4.2.1.1.7

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

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

Step 6.4.2.1.1.8

Multiply $- 1$ by $1$ .

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

Step 6.4.2.1.2

Move the negative in front of the fraction.

$f (\frac{π}{8}) = \frac{1}{2} + 1$

Step 6.4.2.1.3

Multiply $- - \frac{1}{2}$ .

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Step 6.4.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.4.2.1.3.2

Multiply $\frac{1}{2}$ by $1$ .

$f (\frac{π}{8}) = \frac{1}{2} + 1$

Step 6.4.2.2

Simplify the expression.

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Step 6.4.2.2.1

Write $1$ as a fraction with a common denominator.

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

Step 6.4.2.2.2

Combine the numerators over the common denominator.

$f (\frac{π}{8}) = \frac{1 + 2}{2}$

Step 6.4.2.2.3

Add $1$ and $2$ .

$f (\frac{π}{8}) = \frac{3}{2}$

Step 6.4.2.3

The final answer is $\frac{3}{2}$ .

$\frac{3}{2}$

Step 6.5

Find the point at $x = \frac{π}{4}$ .

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Step 6.5.1

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

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

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Simplify the numerator.

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Step 6.5.2.1.1.1

Cancel the common factor of $4$ .

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Step 6.5.2.1.1.1.1

Cancel the common factor.

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

Step 6.5.2.1.1.1.2

Rewrite the expression.

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

Step 6.5.2.1.1.2

Add $π$ and $π$ .

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

Step 6.5.2.1.1.3

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

$f (\frac{π}{4}) = - \frac{\sin (0)}{2} + 1$

Step 6.5.2.1.1.4

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

$f (\frac{π}{4}) = - \frac{0}{2} + 1$

Step 6.5.2.1.2

Divide $0$ by $2$ .

$f (\frac{π}{4}) = - 0 + 1$

Step 6.5.2.1.3

Multiply $- 1$ by $0$ .

$f (\frac{π}{4}) = 0 + 1$

Step 6.5.2.2

Add $0$ and $1$ .

$f (\frac{π}{4}) = 1$

Step 6.5.2.3

The final answer is $1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{4} & 1 \\ - \frac{π}{8} & \frac{1}{2} \\ 0 & 1 \\ \frac{π}{8} & \frac{3}{2} \\ \frac{π}{4} & 1 \end{array}$

Step 7

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

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

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: $1$

$\begin{array}{c} x & f (x) \\ - \frac{π}{4} & 1 \\ - \frac{π}{8} & \frac{1}{2} \\ 0 & 1 \\ \frac{π}{8} & \frac{3}{2} \\ \frac{π}{4} & 1 \end{array}$

Step 8