Find Amplitude, Period, and Phase Shift y=1/2-1/2cos(3x-pi/4)

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

Rewrite the expression as $- \frac{\cos (3 x - \frac{π}{4})}{2} + \frac{1}{2}$ .

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

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = - \frac{1}{2}$

$b = 3$

$c = \frac{π}{4}$

$d = \frac{1}{2}$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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

Find the period of $- \frac{\cos (3 x - \frac{π}{4})}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Find the period of $\frac{1}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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

$\frac{2 π}{3}$

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: $\frac{\frac{π}{4}}{3}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{4} \cdot \frac{1}{3}$

Multiply $\frac{π}{4} \cdot \frac{1}{3}$ .

Multiply $\frac{π}{4}$ by $\frac{1}{3}$ .

Phase Shift: $\frac{π}{4 \cdot 3}$

Multiply $4$ by $3$ .

Phase Shift: $\frac{π}{12}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{12}$ ( $\frac{π}{12}$ to the right)

Vertical Shift: $\frac{1}{2}$