Find Amplitude, Period, and Phase Shift f(x)=-4sin(2x+pi)-5

$f (x) = - 4 \sin (2 x + π) - 5$

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

$a = - 4$

$b = 2$

$c = - π$

$d = - 5$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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

Period: $\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

Period: $\frac{2 π}{| 2 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

Period: $\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Period: $\frac{2 π}{2}$

Divide $π$ by $1$ .

Period: $π$

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: $- \frac{π}{2}$

Find the vertical shift $d$ .

Vertical Shift: $- 5$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $π$

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

Vertical Shift: $- 5$

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Name

Name	one billion sixteen million four hundred thirty-five thousand four hundred eighty-eight

Interesting facts

1016435488 has 128 divisors, whose sum is 7816279680
The reverse of 1016435488 is 8845346101
Previous prime number is 79

Basic properties

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

Name

Name	eight hundred thirty-six million five hundred seventy-one thousand two hundred twenty-one

Interesting facts

836571221 has 4 divisors, whose sum is 859181292
The reverse of 836571221 is 122175638
Previous prime number is 37

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 35
Digital Root 8

Name

Name	one billion one hundred eighty-two million two hundred eighteen thousand one hundred eighty-two

Interesting facts

1182218182 has 8 divisors, whose sum is 1877640696
The reverse of 1182218182 is 2818122811
Previous prime number is 17

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 34
Digital Root 7

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Verify the Identity csc(x)^4-csc(x)^2=cot(x)^4+cot(x)^2

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