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

$y = - 2 \cos (2 x + π) - 1$

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

$a = - 2$

$b = 2$

$c = - π$

$d = - 1$

Find the amplitude $| a |$ .

Amplitude: $2$

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

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

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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 $0$ and $2$ 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 period of $- 1$ .

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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 $0$ and $2$ 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 $1$ .

$π$

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

$π$

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: $\frac{- π}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $π$

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

Vertical Shift: $- 1$

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Name

Name	one billion two hundred seventy-one million five hundred forty-eight thousand three hundred sixty-three

Interesting facts

1271548363 has 4 divisors, whose sum is 1276496280
The reverse of 1271548363 is 3638451721
Previous prime number is 257

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 40
Digital Root 4

Name

Name	six hundred five million two hundred forty-six thousand five hundred fifty-five

Interesting facts

605246555 has 16 divisors, whose sum is 750276864
The reverse of 605246555 is 555642506
Previous prime number is 37

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 38
Digital Root 2

Name

Name	one billion one hundred ninety million three hundred thirty-nine thousand eight hundred seventy-nine

Interesting facts

1190339879 has 4 divisors, whose sum is 1212799176
The reverse of 1190339879 is 9789330911
Previous prime number is 53

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 50
Digital Root 5

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