Find Amplitude, Period, and Phase Shift y=-4cos(pi/3x)

$y = - 4 \cos (\frac{π}{3} x)$

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.

$a = - 4$

$b = \frac{π}{3}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

Step 4

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

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: $\frac{0}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{3}{π})$

Multiply $0$ by $\frac{3}{π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $6$

Phase Shift: None

Vertical Shift: None

Step 6

Find Amplitude, Period, and Phase Shift y=-4cos(pi/3x)

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