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

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

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

$a = - 4$

$b = \frac{1}{3}$

$c = 0$

$d = - 1$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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Find the period of $- 4 \sin (\frac{x}{3})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

Find the period of $- 1$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

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

$6 π$

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: $\frac{0}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 3$

Multiply $0$ by $3$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $6 π$

Phase Shift: None

Vertical Shift: $- 1$

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Name

Name	two hundred twenty-five million nine hundred seventy-nine thousand two hundred sixty

Interesting facts

225979260 has 64 divisors, whose sum is 876107232
The reverse of 225979260 is 062979522
Previous prime number is 13

Basic properties

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

Name

Name	one billion eight hundred sixty-three million three hundred seventy-six thousand six hundred fifty-four

Interesting facts

1863376654 has 4 divisors, whose sum is 2795064984
The reverse of 1863376654 is 4566733681
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 49
Digital Root 4

Name

Name	one billion seven hundred eighty-seven million four hundred fifty thousand two hundred eighty-three

Interesting facts

1787450283 has 8 divisors, whose sum is 3177689408
The reverse of 1787450283 is 3820547871
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 45
Digital Root 9

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