Graph y=2sec(x+pi/4)+1

Graph y=2sec(x+pi/4)+1
Find the asymptotes.
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For any , vertical asymptotes occur at , where is an integer. Use the basic period for , , to find the vertical asymptotes for . Set the inside of the secant function, , for equal to to find where the vertical asymptote occurs for .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Set the inside of the secant function equal to .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
The basic period for will occur at , where and are vertical asymptotes.
Find the period to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.
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The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The vertical asymptotes for occur at , , and every , where is an integer. This is half of the period.
Secant only has vertical asymptotes.
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: where is an integer
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Since the graph of the function does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
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 .
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:
Divide by .
Phase Shift:
Phase Shift:
Find the vertical shift .
Vertical Shift:
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: ( to the left)
Vertical Shift:
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: where is an integer
Amplitude: None
Period:
Phase Shift: ( to the left)
Vertical Shift:
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Name

Name one billion three hundred two million eight hundred twenty-six thousand five hundred eighty-two

Interesting facts

  • 1302826582 has 8 divisors, whose sum is 2039206896
  • The reverse of 1302826582 is 2856282031
  • Previous prime number is 23

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 37
  • Digital Root 1

Name

Name one billion nine hundred ninety-two million one hundred forty-eight thousand nine hundred thirty-one

Interesting facts

  • 1992148931 has 4 divisors, whose sum is 2045990832
  • The reverse of 1992148931 is 1398412991
  • Previous prime number is 37

Basic properties

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

Name

Name one hundred seventy-seven million five hundred twelve thousand nine hundred thirty-seven

Interesting facts

  • 177512937 has 32 divisors, whose sum is 315727872
  • The reverse of 177512937 is 739215771
  • Previous prime number is 47

Basic properties

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