Find Amplitude, Period, and Phase Shift y=5sec(4x)+10

$y = 5 \sec (4 x) + 10$

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

$a = 5$

$b = 4$

$c = 0$

$d = 10$

Since the graph of the function $s e c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

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

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Find the period of $5 \sec (4 x)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

Find the period of $10$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

$\frac{π}{2}$

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

$\frac{π}{2}$

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: $\frac{0}{4}$

Divide $0$ by $4$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{2}$

Phase Shift: None

Vertical Shift: $10$

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Name

Name	one billion seven hundred fifty-three million seven hundred sixteen thousand twenty-one

Interesting facts

1753716021 has 16 divisors, whose sum is 2347583424
The reverse of 1753716021 is 1206173571
Previous prime number is 997

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 33
Digital Root 6

Name

Name	two billion eighty-eight million one hundred ninety-one thousand five hundred forty-five

Interesting facts

2088191545 has 8 divisors, whose sum is 2199476160
The reverse of 2088191545 is 5451918802
Previous prime number is 29

Basic properties

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

Name

Name	one billion four hundred thirteen million forty-three thousand three hundred fifty-two

Interesting facts

1413043352 has 64 divisors, whose sum is 5583253536
The reverse of 1413043352 is 2533403141
Previous prime number is 41

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 26
Digital Root 8

Find Amplitude, Period, and Phase Shift y=3cos(0.5(x+pi))

Find the Foci ((y-2)^2)/64-((x-4)^2)/36=1

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Simplify cos(x)^2csc(x)-csc(x)

Find the Value Using the Unit Circle cos((2pi)/9)