Find Amplitude, Period, and Phase Shift y=1/2sin(1/2x+pi/2)

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

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

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

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

$c = - \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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: $\frac{- \frac{π}{2}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot 2$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{π}{2}$ into the numerator.

Phase Shift: $\frac{- π}{2} \cdot 2$

Cancel the common factor.

Phase Shift: $\frac{- π}{2} \cdot 2$

Rewrite the expression.

Phase Shift: $- π$

List the properties of the trigonometric function.

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

Period: $4 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: None

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Name

Name	one billion five hundred seventy million two hundred seven thousand seven hundred forty-four

Interesting facts

1570207744 has 16384 divisors, whose sum is 177407760384
The reverse of 1570207744 is 4477020751
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 37
Digital Root 1

Name

Name	one hundred eleven million six hundred forty thousand seven hundred twelve

Interesting facts

111640712 has 32 divisors, whose sum is 393170112
The reverse of 111640712 is 217046111
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 23
Digital Root 5

Name

Name	one billion three hundred five million one hundred eighteen thousand six

Interesting facts

1305118006 has 8 divisors, whose sum is 1957830840
The reverse of 1305118006 is 6008115031
Previous prime number is 23459

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 25
Digital Root 7

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