Graph y=-4sec(pix)+2

Graph y=-4sec(pix)+2
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 .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
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.
Set the inside of the secant function equal to .
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
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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is approximately which is positive so remove the absolute value
Cancel the common factor of .
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Cancel the common factor.
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 using the formula .
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Find the period of .
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The period of the function can be calculated using .
Replace with in the formula for period.
is approximately which is positive so remove the absolute value
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Find the period of .
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The period of the function can be calculated using .
Replace with in the formula for period.
is approximately which is positive so remove the absolute value
Cancel the common factor of .
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Cancel the common factor.
Divide by .
The period of addition/subtraction of trig functions is the maximum of the individual periods.
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:
List the properties of the trigonometric function.
Amplitude: None
Period:
Phase Shift: None
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: None
Vertical Shift:
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Name

Name nine hundred seventy-seven million three hundred twenty-nine thousand seven hundred fifty-three

Interesting facts

  • 977329753 has 4 divisors, whose sum is 980052480
  • The reverse of 977329753 is 357923779
  • Previous prime number is 359

Basic properties

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

Name

Name two billion forty-four million eight hundred forty-seven thousand four hundred seventy

Interesting facts

  • 2044847470 has 8 divisors, whose sum is 3680725464
  • The reverse of 2044847470 is 0747484402
  • Previous prime number is 5

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 40
  • Digital Root 4

Name

Name five hundred five million two hundred twenty-eight thousand three hundred eighty-two

Interesting facts

  • 505228382 has 4 divisors, whose sum is 757842576
  • The reverse of 505228382 is 283822505
  • Previous prime number is 2

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 35
  • Digital Root 8