Find Amplitude, Period, and Phase Shift y=4+3sin(pi/3x-pi/16)

$y = 4 + 3 \sin (\frac{π}{3} x - \frac{π}{16})$

Rewrite the expression as $3 \sin (\frac{π x}{3} - \frac{π}{16}) + 4$ .

$3 \sin (\frac{π x}{3} - \frac{π}{16}) + 4$

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

$a = 3$

$b = \frac{π}{3}$

$c = \frac{π}{16}$

$d = 4$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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Find the period of $3 \sin (\frac{π x}{3} - \frac{π}{16})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

Find the period of $4$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{π}{3} |}$

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Rewrite the expression.

$2 \cdot 3$

Multiply $2$ by $3$ .

$6$

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

$6$

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: $\frac{\frac{π}{16}}{\frac{π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{16} \cdot \frac{3}{π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

Phase Shift: $\frac{π}{16} \cdot \frac{3}{π}$

Rewrite the expression.

Phase Shift: $\frac{1}{16} \cdot 3$

Combine $\frac{1}{16}$ and $3$ .

Phase Shift: $\frac{3}{16}$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $6$

Phase Shift: $\frac{3}{16}$ ( $\frac{3}{16}$ to the right)

Vertical Shift: $4$

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Name

Name	one billion eight hundred sixty-eight million seven hundred five thousand eight hundred eighty-seven

Interesting facts

1868705887 has 4 divisors, whose sum is 1886170536
The reverse of 1868705887 is 7885078681
Previous prime number is 107

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 58
Digital Root 4

Name

Name	two billion forty million two hundred eighteen thousand forty-eight

Interesting facts

2040218048 has 512 divisors, whose sum is 25481174400
The reverse of 2040218048 is 8408120402
Previous prime number is 199

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 29
Digital Root 2

Name

Name	one hundred sixty-one million six hundred ninety-two thousand eight hundred seventy-two

Interesting facts

161692872 has 128 divisors, whose sum is 798050880
The reverse of 161692872 is 278296161
Previous prime number is 197

Basic properties

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

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