Find Amplitude, Period, and Phase Shift y=2cos(pi/2x+pi/6)

$y = 2 \cos (\frac{π}{2} x + \frac{π}{6})$

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

$a = 2$

$b = \frac{π}{2}$

$c = - \frac{π}{6}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $2 \cos (\frac{π x}{2} + \frac{π}{6})$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{2} |}$

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Rewrite the expression.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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: $\frac{- \frac{π}{6}}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot \frac{2}{π}$

Cancel the common factor of $π$ .

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Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot \frac{2}{π}$

Factor $π$ out of $- π$ .

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{2}{π}$

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{2}{π}$

Rewrite the expression.

Phase Shift: $\frac{- 1}{6} \cdot 2$

Cancel the common factor of $2$ .

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Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{- 1}{3}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{3}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $4$

Phase Shift: $- \frac{1}{3}$ ( $\frac{1}{3}$ to the left)

Vertical Shift: None

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Name

Name	nine hundred thirty-three million eight hundred seventy thousand nine hundred sixty-seven

Interesting facts

933870967 has 4 divisors, whose sum is 933988896
The reverse of 933870967 is 769078339
Previous prime number is 109391

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 52
Digital Root 7

Name

Name	thirty-five million sixty-nine thousand one hundred fifty-six

Interesting facts

35069156 has 8 divisors, whose sum is 78905610
The reverse of 35069156 is 65196053
Previous prime number is 2

Basic properties

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

Name

Name	five hundred seventy-nine million five hundred seventy-one thousand four hundred

Interesting facts

579571400 has 64 divisors, whose sum is 2036824920
The reverse of 579571400 is 004175975
Previous prime number is 1213

Basic properties

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

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