Find Amplitude, Period, and Phase Shift f(x)=1/4cos(x+pi/3)-1

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

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.

$a = \frac{1}{4}$

$b = 1$

$c = - \frac{π}{3}$

$d = - 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Step 3

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Find the period of $- 1$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$2 π$

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: $\frac{- \frac{π}{3}}{1}$

Divide $- \frac{π}{3}$ by $1$ .

Phase Shift: $- \frac{π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $2 π$

Phase Shift: $- \frac{π}{3}$ ( $\frac{π}{3}$ to the left)

Vertical Shift: $- 1$

Step 6

Find Amplitude, Period, and Phase Shift f(x)=1/4cos(x+pi/3)-1

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