Find Amplitude, Period, and Phase Shift y=2cos(1/2x)+1

$y = 2 \cos (\frac{1}{2} x) + 1$

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

$a = 2$

$b = \frac{1}{2}$

$c = 0$

$d = 1$

Find the amplitude $| a |$ .

Amplitude: $2$

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

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Find the period of $2 \cos (\frac{x}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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

$4 π$

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: $\frac{0}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 2$

Multiply $0$ by $2$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $4 π$

Phase Shift: None

Vertical Shift: $1$

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Name

Name	one billion seven hundred sixty-three million fifty-nine thousand one hundred two

Interesting facts

1763059102 has 64 divisors, whose sum is 3620275200
The reverse of 1763059102 is 2019503671
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 34
Digital Root 7

Name

Name	one hundred ninety-two million three hundred seventy-eight thousand one hundred sixty-four

Interesting facts

192378164 has 32 divisors, whose sum is 492734016
The reverse of 192378164 is 461873291
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 41
Digital Root 5

Name

Name	six hundred seventy-five million seven hundred sixty-five thousand one hundred eighty-nine

Interesting facts

675765189 has 16 divisors, whose sum is 1310575104
The reverse of 675765189 is 981567576
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 54
Digital Root 9

Graph 500 degrees

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Find the Reference Angle -580 degrees

Convert from Radians to Degrees -2p

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