Calculus Examples

$y = - \frac{1}{3} \cos (\frac{1}{3} x)$

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.

$a = - \frac{1}{3}$

$b = \frac{1}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{3}$

Step 3

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 3$

Multiply $0$ by $3$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{3}$

Period: $6 π$

Phase Shift: None

Vertical Shift: None

Step 6