Precalculus Examples

$f (x) = - 4 \sin (8 x + 4) + 6$

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.

$a = - 4$

$b = 8$

$c = - 4$

$d = 6$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

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Step 3.1

Find the period of $- 4 \sin (8 x + 4)$ .

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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 $8$ in the formula for period.

$\frac{2 π}{| 8 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{2 π}{8}$

Step 3.1.4

Cancel the common factor of $2$ and $8$ .

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Step 3.1.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8}$

Step 3.1.4.2

Cancel the common factors.

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Step 3.1.4.2.1

Factor $2$ out of $8$ .

$\frac{2 π}{2 \cdot 4}$

Step 3.1.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 4}$

Step 3.1.4.2.3

Rewrite the expression.

$\frac{π}{4}$

Step 3.2

Find the period of $6$ .

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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 $8$ in the formula for period.

$\frac{2 π}{| 8 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$\frac{2 π}{8}$

Step 3.2.4

Cancel the common factor of $2$ and $8$ .

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Step 3.2.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8}$

Step 3.2.4.2

Cancel the common factors.

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Step 3.2.4.2.1

Factor $2$ out of $8$ .

$\frac{2 π}{2 \cdot 4}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 4}$

Step 3.2.4.2.3

Rewrite the expression.

$\frac{π}{4}$

Step 3.3

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

$\frac{π}{4}$

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: $\frac{- 4}{8}$

Step 4.3

Cancel the common factor of $- 4$ and $8$ .

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Step 4.3.1

Factor $4$ out of $- 4$ .

Phase Shift: $\frac{4 (- 1)}{8}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $4$ out of $8$ .

Phase Shift: $\frac{4 \cdot - 1}{4 \cdot 2}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{4 \cdot - 1}{4 \cdot 2}$

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{π}{4}$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: $6$

Step 6