"Life is like a secant/ cosecant; it has its ups and downs."

WhatIsSecantAndCosecant.jpg

  • Cosecant: 1 / sin x
  • Secant: 1 / cos x

BasicFormOfSecantAndCosecant.jpg

  • y = csc x
  • y = sec x

ThingsToLookForWhenGraphingSecantsAndCosecants.jpg

  • phase shift - can be found by dividing the number which is subtracted or added to x by the number which is multiplied by x
  • amplitude - can be found by looking at the number which is multiplied to the equation
  • inverting of the graph - if the equation is negative the graph is inverted
  • period change - can be found by dividing (2 * Pi) by the number which is multiplied to x
  • vertical shift (occurs rarely) - can be found by looking at the number which is added or subtracted to the equation

Tips.jpg

For Cosecant

  1. Graph the cosecant as if it were a sine (red)
  2. Draw asymptotes where the graph crosses the x-axis (gray)
  3. From the maximums and minimums draw parabolas, bounded by the asymptotes, going to positive and negative infinity (purple)
  4. Erase the sine graph
cosecant.gif

For Secant

  1. Graph the secant as if it were a cosine (blue)
  2. Draw the asymptotes where the graph crosses the x-axis (gray)
  3. From the maximums and minimums draw parabolas, bounded by the asymptotes, going to positive and negative infinity (green)
  4. Erase the cosine graph
secant.gif

PracticeProblems.jpg

  • y = csc x basic
  • y = sec x basic
  • y = csc (x + (Pi / 2)) phase shift
  • y = sec (x - (Pi / 2)) phase shift
  • y = (1 / 2) csc x amplitude
  • y = (1 / 2) sec x amplitude
  • y = -csc x inverting
  • y = -sec x inverting
  • y = csc 2x period change
  • y = sec 2x period change
For further review try problems # 1, 4, 6, 11-18, 23-26, 31-36, 39-41, 43 on page 305 in the text book

Answers.jpg

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Note: csc = cosecant and sec = secant

What You Know About Math?

Note: The following is a youtube video which can not be viewed in school


Ryan Altus and Tom Kusturiss



SECANT GRAPHS:

1. Equation - y = A sec (Bx - C)
  • A = amplitude; in secant graphs, amplitude is undefined
  • Period = 2pi/B
  • Phase shift = C/B
note: this is a general example and NOT the graph of 3 sec (2x - pi)
note: this is a general example and NOT the graph of 3 sec (2x - pi)

How to graph a secant graph:
  1. Graph the COSINE version of the graph with a dotted line.
  2. Graph the asymptotes for your SECANT graph where ever the COSINE graph crosses X-Axis.
  3. The minimum and maximum points of your SECANT graph will be the opposite minimum and maximum points of that COSINE graph; so that the SECANT graph increases or decreases away from the X-Axis.

Example:

y = 3 sec (2x - pi)

  1. Graph the COSINE version (blue) of this graph: y = 3 cos (2x - pi). Amplitude = 2, Period = 2pi/2 = pi, Phase Shift = pi/2.
1st_steph_sec_graph.jpg

2. Now establish the asymptotes (black) of the SECANT graph where the COSINE graph crosses X-Axis. (every pi/2)
2nd_steph_sec_graph.jpg
3. Graph SECANT (pink) function using the minimum and maximum of the COSINE graph as the opposite minium and maximum of SECANT graph.
full_sec_graph.jpg



COSECANT GRAPHS:

1. Equation: A cos (Bx - C)
  • A = amplitude; in secant graphs, amplitude is undefined
  • Period = 2pi/B
  • Phase shift = C/B

How to graph a cosecant graph:
  1. Graph the SINE version of the graph with a dotted line.
  2. Graph the asymptotes for your COSECANT graph where ever the SINE graph crosses X-Axis.
  3. The minimum and maximum points of your COSECANT graph will be the opposite minimum and maximum points of that SINE graph; so that the COSECANT graph increases or decreases away from the X-Axis.
Example:

y = 3 csc (2x - pi)

  1. Graph the SINE version of this graph: y = 3 sin (2x - pi). Amplitude = 2, Period = 2pi/2 = pi, Phase Shift = pi/2.
  2. Now establish the asymptotes of the COSECANT graph where the SINE graph crosses X-Axis. (every pi/2)
  3. Graph COSECANT function using the minimum and maximum of the SINE graph as the opposite minium and maximum of COSECANT graph.
note: this is just a general example and NOT the graph of 3 csc (2x - pi)
note: this is just a general example and NOT the graph of 3 csc (2x - pi)