Find Amplitude, Period, and Phase Shift y=1/4sin(8pix-(5pi)/4)

$y = \frac{1}{4} \sin (8 π x - \frac{5 π}{4})$

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

$a = \frac{1}{4}$

$b = 8 π$

$c = \frac{5 π}{4}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Find the period of $\frac{\sin (8 π x - \frac{5 π}{4})}{4}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $8 π$ in the formula for period.

$\frac{2 π}{| 8 π |}$

$8 π$ is approximately $25.13274122$ which is positive so remove the absolute value

$\frac{2 π}{8 π}$

Cancel the common factor of $2$ and $8$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{8 π}$

Cancel the common factors.

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Factor $2$ out of $8 π$ .

$\frac{2 (π)}{2 (4 π)}$

Cancel the common factor.

$\frac{2 π}{2 (4 π)}$

Rewrite the expression.

$\frac{π}{4 π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{π}{4 π}$

Rewrite the expression.

$\frac{1}{4}$

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: $\frac{\frac{5 π}{4}}{8 π}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{5 π}{4} \cdot \frac{1}{8 π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $5 π$ .

Phase Shift: $\frac{π \cdot 5}{4} \cdot \frac{1}{8 π}$

Factor $π$ out of $8 π$ .

Phase Shift: $\frac{π \cdot 5}{4} \cdot \frac{1}{π \cdot 8}$

Cancel the common factor.

Phase Shift: $\frac{π \cdot 5}{4} \cdot \frac{1}{π \cdot 8}$

Rewrite the expression.

Phase Shift: $\frac{5}{4} \cdot \frac{1}{8}$

Multiply $\frac{5}{4}$ and $\frac{1}{8}$ .

Phase Shift: $\frac{5}{4 \cdot 8}$

Multiply $4$ by $8$ .

Phase Shift: $\frac{5}{32}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $\frac{1}{4}$

Phase Shift: $\frac{5}{32}$ ( $\frac{5}{32}$ to the right)

Vertical Shift: None

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Name

Name	seven hundred fifty-eight million nine hundred forty-seven thousand eight hundred

Interesting facts

758947800 has 128 divisors, whose sum is 3783101760
The reverse of 758947800 is 008749857
Previous prime number is 5

Basic properties

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

Name

Name	one billion six hundred thirty-eight million six hundred fifty-four thousand sixty-nine

Interesting facts

1638654069 has 16 divisors, whose sum is 2476779200
The reverse of 1638654069 is 9604568361
Previous prime number is 19

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 48
Digital Root 3

Name

Name	one hundred twenty-seven million seven hundred twenty-seven thousand two hundred ninety-four

Interesting facts

127727294 has 8 divisors, whose sum is 195667488
The reverse of 127727294 is 492727721
Previous prime number is 47

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 41
Digital Root 5

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