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

$y = \cos (- 2 x + π) - 1$

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

$a = 1$

$b = - 2$

$c = - π$

$d = - 1$

Find the amplitude $| a |$ .

Amplitude: $1$

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

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

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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 $- 2$ 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 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 $- 2$ 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 $1$ .

$π$

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: $\frac{- π}{- 2}$

Dividing two negative values results in a positive value.

Phase Shift: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $π$

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

Vertical Shift: $- 1$

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Name

Name	nine hundred thirty-one million one hundred sixteen thousand eight hundred sixty-three

Interesting facts

931116863 has 4 divisors, whose sum is 931969848
The reverse of 931116863 is 368611139
Previous prime number is 1093

Basic properties

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

Name

Name	two hundred forty-two million ninety-six thousand seven hundred eighty-nine

Interesting facts

242096789 has 4 divisors, whose sum is 264105600
The reverse of 242096789 is 987690242
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 47
Digital Root 2

Name

Name	eight hundred nine million five hundred ninety-three thousand eight hundred thirty-four

Interesting facts

809593834 has 8 divisors, whose sum is 1387875168
The reverse of 809593834 is 438395908
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 49
Digital Root 4

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