Precalculus Examples

$f (x) = 3 \cos (\frac{1}{4} \cdot (π x) + π) - 2$

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.

$a = 3$

$b = \frac{π}{4}$

$c = - π$

$d = - 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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Step 3.1

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

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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 $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Step 3.1.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 3.1.5

Cancel the common factor of $π$ .

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Step 3.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 3.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 3.1.5.3

Rewrite the expression.

$2 \cdot 4$

Step 3.1.6

Multiply $2$ by $4$ .

$8$

Step 3.2

Find the period of $- 2$ .

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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 $\frac{π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{π}{4} |}$

Step 3.2.3

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Step 3.2.5

Cancel the common factor of $π$ .

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Step 3.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Step 3.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Step 3.2.5.3

Rewrite the expression.

$2 \cdot 4$

Step 3.2.6

Multiply $2$ by $4$ .

$8$

Step 3.3

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

$8$

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: $\frac{- π}{\frac{π}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- π \frac{4}{π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Factor $π$ out of $- π$ .

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

Step 4.4.2

Cancel the common factor.

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

Step 4.4.3

Rewrite the expression.

Phase Shift: $- 1 \cdot 4$

Step 4.5

Multiply $- 1$ by $4$ .

Phase Shift: $- 4$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $8$

Phase Shift: $- 4$ ( $4$ to the left)

Vertical Shift: $- 2$

Step 6