Find Amplitude, Period, and Phase Shift y=4-3sin(2/5(x+1))

$y = 4 - 3 \sin (\frac{2}{5} (x + 1))$

Rewrite the expression as $- 3 \sin (\frac{2 x}{5} + \frac{2}{5}) + 4$ .

$- 3 \sin (\frac{2 x}{5} + \frac{2}{5}) + 4$

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

$a = - 3$

$b = \frac{2}{5}$

$c = - \frac{2}{5}$

$d = 4$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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Find the period of $- 3 \sin (\frac{2 x}{5} + \frac{2}{5})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2}{5} |}$

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Cancel the common factor.

$2 π \frac{5}{2}$

Rewrite the expression.

$π \cdot 5$

Move $5$ to the left of $π$ .

$5 π$

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 $\frac{2}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2}{5} |}$

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Cancel the common factor.

$2 π \frac{5}{2}$

Rewrite the expression.

$π \cdot 5$

Move $5$ to the left of $π$ .

$5 π$

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

$5 π$

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: $\frac{- \frac{2}{5}}{\frac{2}{5}}$

Cancel the common factor of $\frac{2}{5}$ .

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Cancel the common factor.

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

Divide $- 1$ by $1$ .

Phase Shift: $- 1$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $5 π$

Phase Shift: $- 1$ ( $1$ to the left)

Vertical Shift: $4$

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Name

Name	seven hundred fifty-one million four hundred ninety-five thousand four hundred twenty-eight

Interesting facts

751495428 has 64 divisors, whose sum is 2585802240
The reverse of 751495428 is 824594157
Previous prime number is 31

Basic properties

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

Name

Name	forty-nine million nine hundred forty-eight thousand six hundred seventeen

Interesting facts

49948617 has 8 divisors, whose sum is 69493824
The reverse of 49948617 is 71684994
Previous prime number is 23

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 48
Digital Root 3

Name

Name	five hundred twenty-five million three hundred ten thousand seven hundred twenty

Interesting facts

525310720 has 2048 divisors, whose sum is 17624945520
The reverse of 525310720 is 027013525
Previous prime number is 11

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 25
Digital Root 7

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