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

$y = 4 \cos (- 2 x - \frac{π}{3})$

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

$a = 4$

$b = - 2$

$c = \frac{π}{3}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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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 $- 2$ and $0$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

Find the phase shift using the formula $\frac{c}{b}$ .

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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: $\frac{\frac{π}{3}}{- 2}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{3} \cdot \frac{1}{- 2}$

Move the negative in front of the fraction.

Phase Shift: $\frac{π}{3} \cdot (- \frac{1}{2})$

Multiply $\frac{π}{3} (- \frac{1}{2})$ .

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Multiply $\frac{π}{3}$ and $\frac{1}{2}$ .

Phase Shift: $- \frac{π}{3 \cdot 2}$

Multiply $3$ by $2$ .

Phase Shift: $- \frac{π}{6}$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: None

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Name

Name	one billion six hundred thirty-one million eight hundred forty-four thousand five hundred eighteen

Interesting facts

1631844518 has 8 divisors, whose sum is 2591753112
The reverse of 1631844518 is 8154481361
Previous prime number is 17

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	eight hundred seventy-eight million seventy-three thousand two hundred thirty-two

Interesting facts

878073232 has 64 divisors, whose sum is 4450329216
The reverse of 878073232 is 232370878
Previous prime number is 887

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 40
Digital Root 4

Name

Name	six hundred two million eight hundred ninety-six thousand six hundred nine

Interesting facts

602896609 has 8 divisors, whose sum is 689308928
The reverse of 602896609 is 906698206
Previous prime number is 2617

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 46
Digital Root 1

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