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

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

Rewrite the expression as $- 4 \sin (\frac{π x}{2} - \frac{π}{6}) - 2$ .

$- 4 \sin (\frac{π x}{2} - \frac{π}{6}) - 2$

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

$a = - 4$

$b = \frac{π}{2}$

$c = \frac{π}{6}$

$d = - 2$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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Find the period of $- 4 \sin (\frac{π x}{2} - \frac{π}{6})$ .

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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 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 $\frac{π}{2}$ 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 $2$ .

$4$

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

$4$

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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Cancel the common factor.

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

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: $\frac{1}{3}$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $4$

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

Vertical Shift: $- 2$

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Name

Name	one billion nine hundred seventy-four million nine hundred fifty-five thousand two hundred sixty-nine

Interesting facts

1974955269 has 8 divisors, whose sum is 2089090432
The reverse of 1974955269 is 9625594791
Previous prime number is 103

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 57
Digital Root 3

Name

Name	one billion eight hundred forty-four million seventy-nine thousand eight hundred thirty-eight

Interesting facts

1844079838 has 8 divisors, whose sum is 2781573480
The reverse of 1844079838 is 8389704481
Previous prime number is 179

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 52
Digital Root 7

Name

Name	five hundred thirty-three million nine hundred fifty-eight thousand eight hundred sixteen

Interesting facts

533958816 has 128 divisors, whose sum is 5406333984
The reverse of 533958816 is 618859335
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 48
Digital Root 3

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