Find Amplitude, Period, and Phase Shift y=8/9cos((pix)/2)

$y = \frac{8}{9} \cos (\frac{π x}{2})$

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

$a = \frac{8}{9}$

$b = \frac{π}{2}$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{8}{9}$

Find the period of $\frac{8 \cos (\frac{π x}{2})}{9}$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{2} |}$

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Rewrite the expression.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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: $\frac{0}{\frac{π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{π})$

Multiply $0$ by $\frac{2}{π}$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $\frac{8}{9}$

Period: $4$

Phase Shift: None

Vertical Shift: None

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Name

Name	thirty-two million four hundred forty-six thousand three hundred thirty-four

Interesting facts

32446334 has 8 divisors, whose sum is 49856688
The reverse of 32446334 is 43364423
Previous prime number is 41

Basic properties

Is Prime? no
Number parity even
Number length 8
Sum of Digits 29
Digital Root 2

Name

Name	one billion one hundred seventy-two million two hundred thirty-nine thousand nine hundred eight

Interesting facts

1172239908 has 32 divisors, whose sum is 4019108544
The reverse of 1172239908 is 8099322711
Previous prime number is 7

Basic properties

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

Name

Name	two hundred forty-six million five hundred four thousand nine

Interesting facts

246504009 has 8 divisors, whose sum is 328744672
The reverse of 246504009 is 900405642
Previous prime number is 9643

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 30
Digital Root 3

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