Find Amplitude, Period, and Phase Shift y=1/2cos((4pix)/5-5pi)

$y = \frac{1}{2} \cos (\frac{4 π x}{5} - 5 π)$

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

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

$b = \frac{4 π}{5}$

$c = 5 π$

$d = 0$

Find the amplitude $| a |$ .

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

Find the period of $\frac{\cos (\frac{4 π x}{5} - 5 π)}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{4 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{4 π}{5} |}$

$\frac{4 π}{5}$ is approximately $2.51327412$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{4 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $4 π$ .

$2 π \frac{5}{2 π (2)}$

Cancel the common factor.

$2 π \frac{5}{2 π \cdot 2}$

Rewrite the expression.

$\frac{5}{2}$

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: $\frac{5 π}{\frac{4 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $5 π (\frac{5}{4 π})$

Cancel the common factor of $π$ .

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Factor $π$ out of $5 π$ .

Phase Shift: $π \cdot (5 (\frac{5}{4 π}))$

Factor $π$ out of $4 π$ .

Phase Shift: $π \cdot (5 (\frac{5}{π \cdot 4}))$

Cancel the common factor.

Phase Shift: $π \cdot (5 (\frac{5}{π \cdot 4}))$

Rewrite the expression.

Phase Shift: $5 (\frac{5}{4})$

Combine $5$ and $\frac{5}{4}$ .

Phase Shift: $\frac{5 \cdot 5}{4}$

Multiply $5$ by $5$ .

Phase Shift: $\frac{25}{4}$

List the properties of the trigonometric function.

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

Period: $\frac{5}{2}$

Phase Shift: $\frac{25}{4}$ ( $\frac{25}{4}$ to the right)

Vertical Shift: None

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Name

Name	one billion three hundred fifteen million two hundred fifty thousand nine hundred seventy-nine

Interesting facts

1315250979 has 16 divisors, whose sum is 2008554240
The reverse of 1315250979 is 9790525131
Previous prime number is 461

Basic properties

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

Name

Name	one billion five hundred forty-two million five hundred fifty-three thousand eight hundred forty-five

Interesting facts

1542553845 has 8 divisors, whose sum is 1558057536
The reverse of 1542553845 is 5483552451
Previous prime number is 3863

Basic properties

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

Name

Name	one billion nine hundred fifteen million seven hundred eighty-seven thousand eight hundred fourteen

Interesting facts

1915787814 has 32 divisors, whose sum is 3864261600
The reverse of 1915787814 is 4187875191
Previous prime number is 397

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 51
Digital Root 6

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