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

$- 3 \cos (2 x + \frac{π}{4})$

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.

$a = - 3$

$b = 2$

$c = - \frac{π}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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: $\frac{- \frac{π}{4}}{2}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{4} \cdot \frac{1}{2}$

Multiply $- \frac{π}{4} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

Phase Shift: $- \frac{π}{2 \cdot 4}$

Multiply $2$ by $4$ .

Phase Shift: $- \frac{π}{8}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $- \frac{π}{8}$ ( $\frac{π}{8}$ to the left)

Vertical Shift: None

Step 6