Graph f(x)=4-2cos(4x+pi)

Graph f(x)=4-2cos(4x+pi)
Rewrite the expression as .
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Find the amplitude .
Amplitude:
Find the period of .
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The period of the function can be calculated using .
Replace with in the formula for period.
The absolute value is the distance between a number and zero. The distance between and is .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the phase shift using the formula .
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The phase shift of the function can be calculated from .
Phase Shift:
Replace the values of and in the equation for phase shift.
Phase Shift:
Move the negative in front of the fraction.
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
Select a few points to graph.
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Find the point at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Add and .
The exact value of is .
Multiply by .
Subtract from .
The final answer is .
Find the point at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Move to the left of .
Add and .
The exact value of is .
Multiply by .
Add and .
The final answer is .
Find the point at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Multiply by .
Add and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
The exact value of is .
Multiply by .
Multiply by .
Add and .
The final answer is .
Find the point at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Move to the left of .
Add and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
Multiply by .
Add and .
The final answer is .
Find the point at .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Add and .
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
Multiply by .
Subtract from .
The final answer is .
List the points in a table.
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
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Name

Name sixty-five million seven hundred sixty-four thousand three hundred twelve

Interesting facts

  • 65764312 has 32 divisors, whose sum is 222685848
  • The reverse of 65764312 is 21346756
  • Previous prime number is 307

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 8
  • Sum of Digits 34
  • Digital Root 7

Name

Name seventy-three million three hundred twenty-nine thousand nine hundred sixty-two

Interesting facts

  • 73329962 has 4 divisors, whose sum is 109994946
  • The reverse of 73329962 is 26992337
  • Previous prime number is 2

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 8
  • Sum of Digits 41
  • Digital Root 5

Name

Name one billion one hundred twenty-four million eight hundred forty-five thousand eight hundred sixty-eight

Interesting facts

  • 1124845868 has 32 divisors, whose sum is 2695853664
  • The reverse of 1124845868 is 8685484211
  • Previous prime number is 167

Basic properties

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