Find Amplitude, Period, and Phase Shift y=3cos(0.5(x+pi))

$y = 3 \cos (0.5 (x + π))$

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

$a = 3$

$b = 0.5$

$c = - 1.57079632$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $3$

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

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

Period: $\frac{2 π}{| b |}$

Replace $b$ with $0.5$ in the formula for period.

Period: $\frac{2 π}{| 0.5 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $0.5$ is $0.5$ .

Period: $\frac{2 π}{0.5}$

Replace $π$ with an approximation.

Period: $\frac{2 \cdot 3.14159265}{0.5}$

Multiply $2$ by $3.14159265$ .

Period: $\frac{6.2831853}{0.5}$

Divide $6.2831853$ by $0.5$ .

Period: $12.56637061$

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: $\frac{- 1.57079632}{0.5}$

Divide $- 1.57079632$ by $0.5$ .

Phase Shift: $- 3.14159265$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $3$

Period: $12.56637061$

Phase Shift: $- 3.14159265$ ( $3.14159265$ to the left)

Vertical Shift: $0$

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Name

Name	fourteen million four hundred eleven thousand one hundred ten

Interesting facts

14411110 has 16 divisors, whose sum is 23203584
The reverse of 14411110 is 01111441
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 8
Sum of Digits 13
Digital Root 4

Name

Name	one billion fifty-nine million seven hundred twenty-four thousand five hundred sixty-four

Interesting facts

1059724564 has 32 divisors, whose sum is 2410948800
The reverse of 1059724564 is 4654279501
Previous prime number is 953

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 43
Digital Root 7

Name

Name	one billion eight hundred forty-two million four hundred twenty-three thousand two hundred seventy-eight

Interesting facts

1842423278 has 4 divisors, whose sum is 2763634920
The reverse of 1842423278 is 8723242481
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

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