Find Amplitude, Period, and Phase Shift y=1/4cos(4x-2pi)

$y = \frac{1}{4} \cos (4 x - 2 π)$

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

$a = \frac{1}{4}$

$b = 4$

$c = 2 π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

Phase Shift: $\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

Phase Shift: $\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $\frac{π}{2}$

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

Vertical Shift: None

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Name

Name	one billion four hundred seventeen million two hundred twenty-three thousand four hundred seventy-four

Interesting facts

1417223474 has 16 divisors, whose sum is 2166652800
The reverse of 1417223474 is 4743227141
Previous prime number is 199

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 35
Digital Root 8

Name

Name	one hundred sixty-eight million five hundred twenty-six thousand three hundred sixty-one

Interesting facts

168526361 has 4 divisors, whose sum is 171382800
The reverse of 168526361 is 163625861
Previous prime number is 59

Basic properties

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

Name

Name	one billion eight hundred thirty-eight million nine hundred fifty-two thousand nine hundred ninety-seven

Interesting facts

1838952997 has 16 divisors, whose sum is 2745026560
The reverse of 1838952997 is 7992598381
Previous prime number is 7

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 61
Digital Root 7

Graph r=9.6

Find the Inverse csc(1.3332)

Evaluate fourth root of 11500

Verify the Identity cos(x-2pi)=cos(x)

Find the Exact Value csc(-780)