Find Amplitude, Period, and Phase Shift y=2sin(4(x-1/2))+3

$y = 2 \sin (4 (x - \frac{1}{2})) + 3$

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

$a = 2$

$b = 4$

$c = 2$

$d = 3$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period using the formula $\frac{2 π}{| b |}$ .

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Find the period of $2 \sin (4 x - 2)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

Find the period of $3$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{π}{2}$

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: $\frac{2}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2$ .

Phase Shift: $\frac{2 (1)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{1}{2}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{π}{2}$

Phase Shift: $\frac{1}{2}$ ( $\frac{1}{2}$ to the right)

Vertical Shift: $3$

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Name

Name	seven hundred eighty-six million nine hundred forty-four thousand nine hundred twenty-eight

Interesting facts

786944928 has 512 divisors, whose sum is 7281135360
The reverse of 786944928 is 829449687
Previous prime number is 29

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 57
Digital Root 3

Name

Name	two hundred twenty-three million five hundred seventy-nine thousand seven hundred twenty-five

Interesting facts

223579725 has 16 divisors, whose sum is 429273216
The reverse of 223579725 is 527975322
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 42
Digital Root 6

Name

Name	one hundred eighty-three million one hundred thirty thousand ninety-five

Interesting facts

183130095 has 16 divisors, whose sum is 202524672
The reverse of 183130095 is 590031381
Previous prime number is 67

Basic properties

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

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