Find Amplitude, Period, and Phase Shift y=1/5cos(1/4x-pi)+2

$y = \frac{1}{5} \cos (\frac{1}{4} x - π) + 2$

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.

$a = \frac{1}{5}$

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

$c = π$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{5}$

Step 3

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

Find the period of $\frac{\cos (\frac{x}{4} - π)}{5}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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

$8 π$

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: $\frac{π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π \cdot 4$

Move $4$ to the left of $π$ .

Phase Shift: $4 π$

Step 5

List the properties of the trigonometric function.

Amplitude: $\frac{1}{5}$

Period: $8 π$

Phase Shift: $4 π$ ( $4 π$ to the right)

Vertical Shift: $2$

Step 6

Find Amplitude, Period, and Phase Shift y=1/5cos(1/4x-pi)+2

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