Find Amplitude, Period, and Phase Shift y=8cos(pi/4x-pi/2)

$y = 8 \cos (\frac{π}{4} x - \frac{π}{2})$

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

$a = 8$

$b = \frac{π}{4}$

$c = \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $8$

Find the period of $8 \cos (\frac{π x}{4} - \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| \frac{π}{4} |}$

$\frac{π}{4}$ is approximately $0.78539816$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{π}$

Cancel the common factor.

$π \cdot 2 \frac{4}{π}$

Rewrite the expression.

$2 \cdot 4$

Multiply $2$ by $4$ .

$8$

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: $\frac{\frac{π}{2}}{\frac{π}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{4}{π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot \frac{4}{π}$

Rewrite the expression.

Phase Shift: $\frac{1}{2} \cdot 4$

Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

Phase Shift: $\frac{1}{2} \cdot (2 (2))$

Cancel the common factor.

Phase Shift: $\frac{1}{2} \cdot (2 \cdot 2)$

Rewrite the expression.

Phase Shift: $2$

List the properties of the trigonometric function.

Amplitude: $8$

Period: $8$

Phase Shift: $2$ ( $2$ to the right)

Vertical Shift: None

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Name

Name	one billion seven hundred eighteen million three hundred three thousand eight hundred thirty-two

Interesting facts

1718303832 has 512 divisors, whose sum is 8465264640
The reverse of 1718303832 is 2383038171
Previous prime number is 47

Basic properties

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

Name

Name	one billion six hundred eighty-three million three hundred forty-three thousand seven hundred ninety-six

Interesting facts

1683343796 has 16 divisors, whose sum is 3788472924
The reverse of 1683343796 is 6973433861
Previous prime number is 4153

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 50
Digital Root 5

Name

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

Interesting facts

1786960598 has 4 divisors, whose sum is 2680440900
The reverse of 1786960598 is 8950696871
Previous prime number is 2

Basic properties

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

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