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

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

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

$a = \frac{2}{3}$

$b = \frac{1}{4}$

$c = \frac{3}{4}$

$d = 1$

Find the amplitude $| a |$ .

Amplitude: $\frac{2}{3}$

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

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Find the period of $\frac{2 \cos (\frac{x - 3}{4})}{3}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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 $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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

$8 π$

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: $\frac{\frac{3}{4}}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3}{4} \cdot 4$

Cancel the common factor of $4$ .

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Cancel the common factor.

Phase Shift: $\frac{3}{4} \cdot 4$

Rewrite the expression.

Phase Shift: $3$

List the properties of the trigonometric function.

Amplitude: $\frac{2}{3}$

Period: $8 π$

Phase Shift: $3$ ( $3$ to the right)

Vertical Shift: $1$

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Name

Name	one billion one hundred sixty-seven million three hundred seventeen thousand two hundred seventy-one

Interesting facts

1167317271 has 8 divisors, whose sum is 1729358960
The reverse of 1167317271 is 1727137611
Previous prime number is 9

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 36
Digital Root 9

Name

Name	one billion seven hundred sixty-nine million sixty-eight thousand five hundred ninety-seven

Interesting facts

1769068597 has 4 divisors, whose sum is 1769998176
The reverse of 1769068597 is 7958609671
Previous prime number is 1907

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 58
Digital Root 4

Name

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

Interesting facts

1650287653 has 16 divisors, whose sum is 1965948544
The reverse of 1650287653 is 3567820561
Previous prime number is 67

Basic properties

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

Find the Quadrant of the Angle 70.9 degrees

Factor sin(x+y)-sin(x-y)

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