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

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

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

$a = 4$

$b = \frac{2}{5}$

$c = 1$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $4$

Find the period of $4 \cos (\frac{2 x}{5} - 1)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2}{5} |}$

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{5}{2}$

Cancel the common factor.

$2 π \frac{5}{2}$

Rewrite the expression.

$π \cdot 5$

Move $5$ to the left of $π$ .

$5 π$

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: $\frac{1}{\frac{2}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $1 (\frac{5}{2})$

Multiply $\frac{5}{2}$ by $1$ .

Phase Shift: $\frac{5}{2}$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $5 π$

Phase Shift: $\frac{5}{2}$ ( $\frac{5}{2}$ to the right)

Vertical Shift: None

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Name

Name	one billion three hundred forty million four hundred thirty-nine thousand three hundred thirty-five

Interesting facts

1340439335 has 8 divisors, whose sum is 1622762640
The reverse of 1340439335 is 5339340431
Previous prime number is 113

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 35
Digital Root 8

Name

Name	one billion seven hundred ninety-seven million three hundred twenty-one thousand four hundred ninety-four

Interesting facts

1797321494 has 4 divisors, whose sum is 2695982244
The reverse of 1797321494 is 4941237971
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 47
Digital Root 2

Name

Name	four hundred forty-six million six hundred fifty-eight thousand seven hundred seventy-three

Interesting facts

446658773 has 16 divisors, whose sum is 495173952
The reverse of 446658773 is 377856644
Previous prime number is 281

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 50
Digital Root 5

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