Find Amplitude, Period, and Phase Shift y=-4cos(5x-9)-7

$y = - 4 \cos (5 x - 9) - 7$

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.

$a = - 4$

$b = 5$

$c = 9$

$d = - 7$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

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

Find the period of $- 4 \cos (5 x - 9)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Find the period of $- 7$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

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

$\frac{2 π}{5}$

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: $\frac{9}{5}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $\frac{2 π}{5}$

Phase Shift: $\frac{9}{5}$ ( $\frac{9}{5}$ to the right)

Vertical Shift: $- 7$

Step 6