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

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

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

$a = 1$

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

$c = - \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

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

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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: $1$

Period: $4 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: None

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Name

Name	one billion three hundred eighty-four million six hundred ninety-nine thousand eight hundred sixty-nine

Interesting facts

1384699869 has 16 divisors, whose sum is 2511928000
The reverse of 1384699869 is 9689964831
Previous prime number is 49

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 63
Digital Root 9

Name

Name	eighty-six million three hundred sixty-five thousand one hundred ninety-four

Interesting facts

86365194 has 32 divisors, whose sum is 179283456
The reverse of 86365194 is 49156368
Previous prime number is 197

Basic properties

Is Prime? no
Number parity even
Number length 8
Sum of Digits 42
Digital Root 6

Name

Name	nine hundred twenty-five million seven hundred thousand three hundred eighty-six

Interesting facts

925700386 has 16 divisors, whose sum is 1496073600
The reverse of 925700386 is 683007529
Previous prime number is 2459

Basic properties

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

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