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

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

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

$a = - 2$

$b = 3$

$c = \frac{3 π}{4}$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $2$

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

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Find the period of $- 2 \sin (3 x - \frac{3 π}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Find the period of $2$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

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

$\frac{2 π}{3}$

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: $\frac{\frac{3 π}{4}}{3}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

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Factor $3$ out of $3 π$ .

Phase Shift: $\frac{3 (π)}{4} \cdot \frac{1}{3}$

Cancel the common factor.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{3}$

Rewrite the expression.

Phase Shift: $\frac{π}{4}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $2$

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Name

Name	three hundred eighty-five million five hundred eleven thousand seven hundred sixty-three

Interesting facts

385511763 has 32 divisors, whose sum is 519782400
The reverse of 385511763 is 367115583
Previous prime number is 7

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 39
Digital Root 3

Name

Name	one billion fifty-seven million six hundred forty-eight thousand three hundred nine

Interesting facts

1057648309 has 4 divisors, whose sum is 1072136716
The reverse of 1057648309 is 9038467501
Previous prime number is 73

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 43
Digital Root 7

Name

Name	ninety-six million sixty thousand seven hundred eighty-seven

Interesting facts

96060787 has 4 divisors, whose sum is 96082360
The reverse of 96060787 is 78706069
Previous prime number is 6283

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 43
Digital Root 7

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