Find Amplitude, Period, and Phase Shift y=1/4sin((4pix)/7+(2pi)/5)

$y = \frac{1}{4} \sin (\frac{4 π x}{7} + \frac{2 π}{5})$

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

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

$b = \frac{4 π}{7}$

$c = - \frac{2 π}{5}$

$d = 0$

Find the amplitude $| a |$ .

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

Find the period of $\frac{\sin (\frac{4 π x}{7} + \frac{2 π}{5})}{4}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{4 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{4 π}{7} |}$

$\frac{4 π}{7}$ is approximately $1.7951958$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4 π}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{4 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $4 π$ .

$2 π \frac{7}{2 π (2)}$

Cancel the common factor.

$2 π \frac{7}{2 π \cdot 2}$

Rewrite the expression.

$\frac{7}{2}$

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: $\frac{- \frac{2 π}{5}}{\frac{4 π}{7}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{2 π}{5} \cdot \frac{7}{4 π}$

Cancel the common factor of $2 π$ .

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Move the leading negative in $- \frac{2 π}{5}$ into the numerator.

Phase Shift: $\frac{- 2 π}{5} \cdot \frac{7}{4 π}$

Factor $2 π$ out of $- 2 π$ .

Phase Shift: $\frac{2 π (- 1)}{5} \cdot \frac{7}{4 π}$

Factor $2 π$ out of $4 π$ .

Phase Shift: $\frac{2 π (- 1)}{5} \cdot \frac{7}{2 π (2)}$

Cancel the common factor.

Phase Shift: $\frac{2 π \cdot - 1}{5} \cdot \frac{7}{2 π \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{- 1}{5} \cdot \frac{7}{2}$

Multiply $\frac{- 1}{5}$ and $\frac{7}{2}$ .

Phase Shift: $\frac{- 1 \cdot 7}{5 \cdot 2}$

Simplify the expression.

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Multiply $- 1$ by $7$ .

Phase Shift: $\frac{- 7}{5 \cdot 2}$

Multiply $5$ by $2$ .

Phase Shift: $\frac{- 7}{10}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{7}{10}$

List the properties of the trigonometric function.

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

Period: $\frac{7}{2}$

Phase Shift: $- \frac{7}{10}$ ( $\frac{7}{10}$ to the left)

Vertical Shift: None

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Name

Name	five hundred sixty-one million nine hundred seven thousand two hundred seventy-eight

Interesting facts

561907278 has 16 divisors, whose sum is 956439360
The reverse of 561907278 is 872709165
Previous prime number is 47

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 45
Digital Root 9

Name

Name	one billion eight hundred million six hundred eighty-four thousand nine hundred ninety-five

Interesting facts

1800684995 has 4 divisors, whose sum is 2160822000
The reverse of 1800684995 is 5994860081
Previous prime number is 5

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 50
Digital Root 5

Name

Name	nine hundred twenty-eight million one hundred twenty-five thousand sixty

Interesting facts

928125060 has 16 divisors, whose sum is 2227500288
The reverse of 928125060 is 060521829
Previous prime number is 15

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 33
Digital Root 6

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