Graph y=cos(2x+pi/4)

Graph y=cos(2x+pi/4)
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 .
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Cancel the common factor.
Divide by .
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:
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Multiply .
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Multiply and .
Phase Shift:
Multiply by .
Phase Shift:
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.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Simplify terms.
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Combine the numerators over the common denominator.
Add and .
Divide by .
The exact value of is .
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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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine the numerators over the common denominator.
Add and .
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.
The exact value of is .
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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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine the numerators over the common denominator.
Add and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
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 .
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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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine the numerators over the common denominator.
Add and .
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.
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
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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Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine the numerators over the common denominator.
Add and .
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.
Divide by .
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
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 one billion four hundred sixty-four million one hundred fifty-one thousand seven hundred ninety-nine

Interesting facts

  • 1464151799 has 8 divisors, whose sum is 1681322880
  • The reverse of 1464151799 is 9971514641
  • Previous prime number is 19

Basic properties

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

Name

Name three hundred forty-eight million eight hundred fifty-four thousand two hundred sixty-one

Interesting facts

  • 348854261 has 8 divisors, whose sum is 399288384
  • The reverse of 348854261 is 162458843
  • Previous prime number is 673

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 41
  • Digital Root 5

Name

Name eight hundred thirty-six million seven hundred forty-nine thousand nine hundred sixty

Interesting facts

  • 836749960 has 32 divisors, whose sum is 3388837500
  • The reverse of 836749960 is 069947638
  • Previous prime number is 5

Basic properties

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