Find Amplitude, Period, and Phase Shift f(x)=1-3sin(2x-9)

$f (x) = 1 - 3 \sin (2 x - 9)$

Rewrite the expression as $- 3 \sin (2 x - 9) + 1$ .

$- 3 \sin (2 x - 9) + 1$

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

$a = - 3$

$b = 2$

$c = 9$

$d = 1$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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Find the period of $- 3 \sin (2 x - 9)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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 $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

$π$

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: $\frac{9}{2}$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $π$

Phase Shift: $\frac{9}{2}$ ( $\frac{9}{2}$ to the right)

Vertical Shift: $1$

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Name

Name	one billion two hundred ninety-two million four hundred four thousand nine hundred ninety-two

Interesting facts

1292404992 has 2048 divisors, whose sum is 44235416736
The reverse of 1292404992 is 2994042921
Previous prime number is 947

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 42
Digital Root 6

Name

Name	one billion three hundred ninety-six million eight hundred sixty-five thousand eight hundred eighty-five

Interesting facts

1396865885 has 8 divisors, whose sum is 1691073720
The reverse of 1396865885 is 5885686931
Previous prime number is 113

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 59
Digital Root 5

Name

Name	seven hundred forty-two million three hundred eight thousand forty-three

Interesting facts

742308043 has 4 divisors, whose sum is 785973240
The reverse of 742308043 is 340803247
Previous prime number is 17

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 31
Digital Root 4

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