Precalculus Examples

$f (x) = - \frac{2}{3} \cos (\frac{π}{4} + 4 x)$

Step 1

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

$a = - \frac{2}{3}$

$b = 4$

$c = - \frac{π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{2}{3}$

Step 3

Find the period of $- \frac{2 \cos (\frac{π}{4} + 4 x)}{3}$ .

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

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.4

Multiply $- \frac{π}{4} \cdot \frac{1}{4}$ .

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Step 4.4.1

Multiply $\frac{1}{4}$ by $\frac{π}{4}$ .

Phase Shift: $- \frac{π}{4 \cdot 4}$

Step 4.4.2

Multiply $4$ by $4$ .

Phase Shift: $- \frac{π}{16}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{2}{3}$

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{16}$ ( $\frac{π}{16}$ to the left)

Vertical Shift: None

Step 6