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

$y = 3 \cos (\frac{2 x}{3} + 3 π)$

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

$a = 3$

$b = \frac{2}{3}$

$c = - 3 π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{2}{3} |}$

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Cancel the common factor.

$2 π \frac{3}{2}$

Rewrite the expression.

$π \cdot 3$

Move $3$ to the left of $π$ .

$3 π$

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: $\frac{- 3 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 3 π \frac{3}{2}$

Multiply $- 3 π \frac{3}{2}$ .

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Combine $\frac{3}{2}$ and $- 3$ .

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

Multiply $3$ by $- 3$ .

Phase Shift: $\frac{- 9}{2} \cdot π$

Combine $\frac{- 9}{2}$ and $π$ .

Phase Shift: $\frac{- 9 π}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{9 π}{2}$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $3 π$

Phase Shift: $- \frac{9 π}{2}$ ( $\frac{9 π}{2}$ to the left)

Vertical Shift: None

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Name

Name	two hundred three million nine hundred thirty-nine thousand seven hundred fifty

Interesting facts

203939750 has 32 divisors, whose sum is 453099744
The reverse of 203939750 is 057939302
Previous prime number is 5

Basic properties

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

Name

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

Interesting facts

1531393056 has 512 divisors, whose sum is 16040566080
The reverse of 1531393056 is 6503931351
Previous prime number is 43

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 36
Digital Root 9

Name

Name	four hundred twenty million two hundred thirty-nine thousand seventy-three

Interesting facts

420239073 has 8 divisors, whose sum is 565759168
The reverse of 420239073 is 370932024
Previous prime number is 103

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 30
Digital Root 3

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