Find Amplitude, Period, and Phase Shift f(x)=-1/2cos(4(x+pi/4))+1

$f (x) = - \frac{1}{2} \cos (4 (x + \frac{π}{4})) + 1$

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

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

$b = 4$

$c = - π$

$d = 1$

Find the amplitude $| a |$ .

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

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

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ 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 $1$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $4$ 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: $\frac{- π}{4}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{4}$

List the properties of the trigonometric function.

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

Period: $\frac{π}{2}$

Phase Shift: $- \frac{π}{4}$ ( $\frac{π}{4}$ to the left)

Vertical Shift: $1$

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Name

Name	one billion seven hundred forty-seven million six hundred sixty-five thousand three hundred twenty-one

Interesting facts

1747665321 has 8 divisors, whose sum is 2356403040
The reverse of 1747665321 is 1235667471
Previous prime number is 89

Basic properties

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

Name

Name	two hundred eighty-three million nine hundred sixty-six thousand nine hundred twenty-seven

Interesting facts

283966927 has 8 divisors, whose sum is 294184800
The reverse of 283966927 is 729669382
Previous prime number is 727

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 52
Digital Root 7

Name

Name	three hundred thirty-five million three hundred twenty-five thousand two hundred forty

Interesting facts

335325240 has 128 divisors, whose sum is 1522501920
The reverse of 335325240 is 042523533
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 27
Digital Root 9

Expand (x+2)^6

Convert from Radians to Degrees -1.5

Graph y=-2sin(3(x-pi/8))

Determine if True f(2/4)=3(2/4)^2-12(2/4)+6

Graph y=5cot(4x)