Find Amplitude, Period, and Phase Shift y=-3cos(pix+2)-6

$y = - 3 \cos (π x + 2) - 6$

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.

$a = - 3$

$b = π$

$c = - 2$

$d = - 6$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

Find the period of $- 6$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

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

$2$

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: $\frac{- 2}{π}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{2}{π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $2$

Phase Shift: $- \frac{2}{π}$ ( $\frac{2}{π}$ to the left)

Vertical Shift: $- 6$

Step 6

Find Amplitude, Period, and Phase Shift y=-3cos(pix+2)-6

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