Graph y=cos(3x)

Graph y=cos(3x)
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 .
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:
Divide by .
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
Vertical Shift:
Select a few points to graph.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Multiply by .
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
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 .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Combine and .
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
The exact value of is .
Find the point at .
Replace the variable with in the expression.
Simplify the result.
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 .
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 right)
Vertical Shift:
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Name

Name four hundred twenty-three million four hundred fifty-five thousand two hundred sixty-nine

Interesting facts

• 423455269 has 4 divisors, whose sum is 434900044
• The reverse of 423455269 is 962554324
• Previous prime number is 37

Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 40
• Digital Root 4

Name

Name four hundred seventy-one million four hundred two thousand one hundred fourteen

Interesting facts

• 471402114 has 16 divisors, whose sum is 947691552
• The reverse of 471402114 is 411204174
• Previous prime number is 193

Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 24
• Digital Root 6

Name

Name one billion three hundred two million fifty-five thousand three hundred ninety-six

Interesting facts

• 1302055396 has 16 divisors, whose sum is 2929972500
• The reverse of 1302055396 is 6935502031
• Previous prime number is 12401

Basic properties

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