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

$f (x) = - 4 - 5 \cos (2 x - π)$

Rewrite the expression as $- 5 \cos (2 x - π) - 4$ .

$- 5 \cos (2 x - π) - 4$

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

$a = - 5$

$b = 2$

$c = π$

$d = - 4$

Find the amplitude $| a |$ .

Amplitude: $5$

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

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Find the period of $- 5 \cos (2 x - π)$ .

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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 $- 4$ .

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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: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $π$

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

Vertical Shift: $- 4$

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Name

Name	three million thirty-three thousand three hundred sixty-three

Interesting facts

3033363 has 8 divisors, whose sum is 4096000
The reverse of 3033363 is 3633303
Previous prime number is 79

Basic properties

Is Prime? no
Number parity odd
Number length 7
Sum of Digits 21
Digital Root 3

Name

Name	six hundred eighty-eight million six hundred seventy-six thousand five hundred sixty-three

Interesting facts

688676563 has 8 divisors, whose sum is 716322816
The reverse of 688676563 is 365676886
Previous prime number is 31

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 55
Digital Root 1

Name

Name	thirty-two million six hundred twenty-six thousand four hundred fifty-three

Interesting facts

32626453 has 4 divisors, whose sum is 32656780
The reverse of 32626453 is 35462623
Previous prime number is 1117

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 31
Digital Root 4

Evaluate arcsin(cos(pi/3))

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Verify the Identity sec(pi/2-x)=csc(x)

Find the Sine (1,2)

Find the Asymptotes y=(3/2)^x