Find Amplitude, Period, and Phase Shift y=cos((2pix)/3+1/3)

$y = \cos (\frac{2 π x}{3} + \frac{1}{3})$

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

$a = 1$

$b = \frac{2 π}{3}$

$c = - \frac{1}{3}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\cos (\frac{2 π x}{3} + \frac{1}{3})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Cancel the common factor of $2 π$ .

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Cancel the common factor.

$2 π \frac{3}{2 π}$

Rewrite the expression.

$3$

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: $\frac{- \frac{1}{3}}{\frac{2 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $- \frac{1}{2 π}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $3$

Phase Shift: $- \frac{1}{2 π}$ ( $\frac{1}{2 π}$ to the left)

Vertical Shift: None

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Name

Name	nine hundred thirty-eight million six hundred two thousand five hundred seventy

Interesting facts

938602570 has 8 divisors, whose sum is 1689484644
The reverse of 938602570 is 075206839
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 40
Digital Root 4

Name

Name	one billion five hundred twelve million thirty-two thousand eight hundred seventy

Interesting facts

1512032870 has 8 divisors, whose sum is 2294732496
The reverse of 1512032870 is 0782302151
Previous prime number is 85

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 29
Digital Root 2

Name

Name	seven hundred fifty-nine million one hundred ninety-six thousand two hundred seven

Interesting facts

759196207 has 8 divisors, whose sum is 946530432
The reverse of 759196207 is 702691957
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 46
Digital Root 1

Graph y=x^2-12x+2

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