Find Amplitude, Period, and Phase Shift y=1/2*cos(4x-(2pi)/3)

$y = \frac{1}{2} \cdot \cos (4 x - \frac{2 π}{3})$

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

$a = \frac{1}{2}$

$b = 4$

$c = \frac{2 π}{3}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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 $4$ in the formula for period.

Period: $\frac{2 π}{| 4 |}$

Solve the equation.

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

Period: $\frac{2 π}{4}$

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

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Factor $2$ out of $2 π$ .

Period: $\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

Period: $\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

Period: $\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

Period: $\frac{π}{2}$

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: $\frac{\frac{2 π}{3}}{4}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{2 π}{3} \cdot \frac{1}{4}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (π)}{3} \cdot \frac{1}{4}$

Factor $2$ out of $4$ .

Phase Shift: $\frac{2 (π)}{3} \cdot \frac{1}{2 (2)}$

Cancel the common factor.

Phase Shift: $\frac{2 π}{3} \cdot \frac{1}{2 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{π}{3} \cdot \frac{1}{2}$

Multiply $\frac{π}{3}$ and $\frac{1}{2}$ .

Phase Shift: $\frac{π}{3 \cdot 2}$

Multiply $3$ by $2$ .

Phase Shift: $\frac{π}{6}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{π}{2}$

Phase Shift: $\frac{π}{6}$ ( $\frac{π}{6}$ to the right)

Vertical Shift: $0$

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Name

Name	one billion two million three hundred thirty-two thousand five hundred

Interesting facts

1002332500 has 128 divisors, whose sum is 3637382112
The reverse of 1002332500 is 0052332001
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 16
Digital Root 7

Name

Name	two billion one hundred thirty million three hundred fifty-three thousand six hundred ninety-two

Interesting facts

2130353692 has 16 divisors, whose sum is 4806149832
The reverse of 2130353692 is 2963530312
Previous prime number is 373

Basic properties

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

Name

Name	three hundred eight million fifteen thousand three hundred sixty-eight

Interesting facts

308015368 has 16 divisors, whose sum is 1039551894
The reverse of 308015368 is 863510803
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 34
Digital Root 7

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