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 .
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 .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Find the phase shift using the formula .
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.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
The exact value of is .
Multiply by .
Subtract from .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Simplify each term.
Cancel the common factor of .
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.
Move to the left of .
The exact value of is .
Multiply by .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Multiply 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 .
Multiply by .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Cancel the common factor of .
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.
Move to the left of .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
Multiply by .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
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 .
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 three hundred ninety-seven million two hundred ninety thousand nine hundred forty-nine

Interesting facts

• 1397290949 has 4 divisors, whose sum is 1420197420
• The reverse of 1397290949 is 9490927931
• Previous prime number is 61

Basic properties

• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 53
• Digital Root 8

Name

Name two hundred sixty-five million one hundred fourteen thousand forty

Interesting facts

• 265114040 has 32 divisors, whose sum is 1073712024
• The reverse of 265114040 is 040411562
• Previous prime number is 5

Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 23
• Digital Root 5

Name

Name one billion four hundred nine million seven hundred three thousand three hundred sixteen

Interesting facts

• 1409703316 has 64 divisors, whose sum is 3688998912
• The reverse of 1409703316 is 6133079041
• Previous prime number is 487

Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 34
• Digital Root 7