Graphs+of+Secant+and+Cosecant+and+their+Translations

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

[[image:WhatIsSecantAndCosecant.jpg]]

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

[[image:BasicFormOfSecantAndCosecant.jpg]]

 * y = csc x
 * y = sec x

[[image:ThingsToLookForWhenGraphingSecantsAndCosecants.jpg width="777" height="80"]]

 * 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

**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

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

[[image:PracticeProblems.jpg]]
For further review try problems # 1, 4, 6, 11-18, 23-26, 31-36, 39-41, 43 on page 305 in the text book
 * 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**

[[image:Answers.jpg]]
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 media type="custom" key="522731"

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**
 * 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.

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

__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.