# Graph y=4cos(2x)+3

Graph y=4cos(2x)+3
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 .
Cancel the common factor.
Divide by .
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.
Simplify each term.
Multiply by .
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 .
Factor out of .
Cancel the common factor.
Rewrite the expression.
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.
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.
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.
Subtract full rotations of until the angle is greater than or equal to and less than .
The exact value of is .
Multiply by .
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 three hundred thirty-one million five hundred ninety-one thousand three hundred eighty-seven

### Interesting facts

• 331591387 has 8 divisors, whose sum is 338672880
• The reverse of 331591387 is 783195133
• Previous prime number is 233

### Basic properties

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

### Name

Name seven hundred fifty-two million six hundred fifty-six thousand seven hundred thirty-one

### Interesting facts

• 752656731 has 8 divisors, whose sum is 1024894464
• The reverse of 752656731 is 137656257
• Previous prime number is 47

### Basic properties

• Is Prime? no
• Number parity odd
• Number length 9
• Sum of Digits 42
• Digital Root 6

### Name

Name nine hundred eighty-eight million three hundred forty thousand fifty-eight

### Interesting facts

• 988340058 has 32 divisors, whose sum is 2638264896
• The reverse of 988340058 is 850043889
• Previous prime number is 997

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 45
• Digital Root 9