Graphing+Trig+Functions

Unit 5 - Graphing Trigonometric Functions



(Emma Welch)
 * __ Original Practice Problems for Unit 5A __**__ : __


 * **// Beginner Problem //**

Give the amplitude, period, frequency, midline, vertical shift, and phase shift of the following formula:

Y= 3 sin(4x-π/2)+2


 * //** On the Way Problem **//

Graph the following equation:

Y= -tan(x+π)


 * **// Got It Problems //**

Write a cosine function for the following graph:



Write a sine function for the following graph:




 * //** Rockstar Problem **//

Write two formulas using two different trigonometric functions for the graph:





(Emma Welch)
 * __Original Practice Problems for Unit 5B__**__:__


 * ** // Beginner Problem // **

Give the amplitude, period, frequency, midline, vertical shift, and phase shift of the following formula:

Y= 4 sec(2x-π/4)-3


 * // ** On the Way Problems ** //

Graph the following equations:

Y= -csc(x/2)+1

Y= 3 sec(2x- π)


 * ** // Got It Problems // **

Write a cotangent function for the following graph:



Write a cosecant function for the following graph:





(Kayla Broxson)
 * __Helpful Websites:__ **

http://www.clarku.edu/~djoyce/trig/functions.html

This site is very informative on the subjects of trigonometric functions. It specifically discusses the definition of sine, cosine, and tangent in relation to the unit circle as well as the characteristics of each function. Also found on the website is a parent graph of each function and an explanation of the individual and shared features. This site also reminds us of the concept concerning positive and negative revolutions around the unit circle and how the sine, cosine, and tangent differ which each example.

http://www.purplemath.com/modules/grphtrig.htm

This website focuses more on the actual graphing of trigonometric functions. The graphs of cosine, sine, and tangent are thoroughly analyzed and broken down. The elements of the trigonometric equations are also explained in depth to viewers. You can also learn about the phase shift, period, frequency, amplitude, and midline on this particular site. If you are having trouble remembering the “a,b,c,d” values in the equation, I would recommend you look at this site to find helpful explanations of said concepts.

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If you are having trouble remembering the specific characteristics of graphs, then this site will be very helpful. It displays and explains the specific features such as: x-intercepts, y-intercepts, periods, as well as an example of a graphed cosecant, secant, tangent, and cotangent function. Furthermore, this site also provides a helpful link with instructions on how to graph these functions in a TI-84 calculator.

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This website also breaks down each secant, cosecant, and cotangent functions by explaining the distinct and similar characteristics of each. They are described in terms of their reciprocals (of which we are already familiar with), so it is much easier to understand. Along with pictures of graphs and plotted points, we are also provided with example problems and practice problems.

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On this site, you can find a clear graph of the basic trig functions. Alongside the graphs, there are definitions and explanations of each. Transformations of the parent graphs are included as well. This site even allows you to quiz yourself as you are reading and studying the information, making it an ideal study tool.

(Stephanie Moreno)
 * __ Jobs and Occupations that use Graphs of Trigonometric Functions __**__ : __

Aerospace Engineers:

Aerospace Engineers perform a variety of engineering work in designing, constructing, testing aircraft, missiles and spacecraft. May conduct basic and applied research adaptability of materials and equipment to aircraft design and manufacture, and may recommend improvements in testing equipment and techniques. Trigonometric graphs are used for modeling many different natural and mechanical phenomena (engines, acoustics, electronics, UV intensity, etc).

http://www.xpmath.com/careers/jobsresult.php?groupID=2&jobID=12

A Day in the Life - Aerospace Engineer: media type="custom" key="25054750"

Electricians:

Electricians work with the systems that provide electricity to homes and businesses. They may install wiring, heating and air-conditioning systems, or make repairs to such systems. They must follow government rules and building codes to ensure their safety and the safety of the buildings they work on. Trigonometric functions and tangent functions can help an electrician correctly bend the conduit of a house into any angle; it offers certain advantages, such as protecting the electric wires from damage, bonding to the ground throughout the entire circuit, and enabling the user to add more circuits later on.

http://www.xpmath.com/careers/jobsresult.php?groupID=7&jobID=10



__**Unit 5A Summary - General Rules and Equations**:__ (Brenden DeArman)


 * y=a sin (bx+c)+d

OR


 * y=a cos (bx+c)+d


 * Period: 2pi/b

//. - __Period__ is the distance at which a wave starts to repeat itself again.//


 * Phase Shift: -c/b


 * Amplitude: a

//. - Making the amplitude __negative__ will cause a reflection across the x-axis.//


 * Vertical Shift: d

__ Sine Graphs: y=sin(x) __


 * CROSSES Y-AXIS AT MIDLINE*


 * Period: 2π
 * Amplitude: 1
 * Midline: y=0
 * X-intercepts: 0, 1π, 2π
 * Max: 1
 * Min: -1

__ Cosine Graphs: y=cos(x) __




 * CROSSES Y-AXIS AT MAX*


 * Period: 2π
 * Amplitude: 1
 * Midline: y=0
 * X-intercepts:π/2, π, 3π/2
 * Max: 1
 * Min: -1

__ Tangent Graphs: y=tan(x) __
 * Period: π
 * Midline: y=0
 * X-intercepts: 0, π, 2π
 * Vertical Asymptotes: π/2, 3π/2

Changing the amplitude of a sine or cosine graph will change the height of the wave. Since tangent doesn’t have waves, changing the amplitude will, in a sense, change the curviness of the line.

__** Annotated Bibliography **:__ (Daniel Reyes)



// __ Trigonometry for Dummies __ //

Sterling, Mary Jane. //Trigonometry for Dummies//. Hoboken, NJ: Wiley, 2005. Print.

__Trigonometry for Dummies__ is part of a series of easy to understand books that provide extra help to many subjects. This book discusses the characteristics of cosine and sine functions and their graphs starting on chapter 16. It initially goes on to hit some of the most basic characteristics of both functions on a graph (sine passing through the origin and cosine intercepting the y-axis at its maximum). The book was very helpful in defining and explaining all the parts that a trigonometric function can have such as amplitude, period, phase shift, and mid-line. __Trigonometry for Dummies__ made it easier for me to understand what is caused by changing each of the parts of a function because it provides you with examples and graphics that are meant for people to further grasp the content. Chapter 17 used the same easy-to-understand methods to explain the characteristics of the parent tangent function. Additionally, the book explains fundaments, such as obtaining the tangent, cosine, and sine of right triangles, which are needed in order to understand the three major trigonometric functions well. In other words, this very useful book was written for anyone in need of an explanation or a clarification for any sub-topic relating to trigonometry.

__** Instructional Videos (5A and 5B) ** : __ (Mandi W.)

http://www.youtube.com/watch?v=R0RM6FeiD8k This video discusses all the parent graphs for sine,cosine,tangent,secant, and cosecant.

http://www.youtube.com/watch?v=80c_F0-7ZxE This video discusses how to find period, amplitude, b, and mid-line/d.

http://www.youtube.com/watch?v=XhDlntep4Sg This video uses a bike wheel and other things such as average rainfall and average high temperatures to show us how to apply trig functions to real life.

http://www.youtube.com/watch?v=UUtYnuStELs This video teaches you how to evaluate inverse trig functions by hand or with a graphing calculator

http://www.youtube.com/watch?v=sESSCAkB2Yg This video shows the inverse trig functions being graphed.

__ Unit 5B Summary __ > View the summary of chapter 5A for all of the skills you need to know entering this section of the chapter. > __ Steps for Graphing Cosecant and Secant: __ > Note: There should be one point of tangency for each U-shape. >
 * Pre-Requisite Skills:
 * Notes:
 * 1) Graph the reciprocal sinusoid. (sine for cosecant and cosine for secant)
 * 2) Sketch the vertical asymptotes wherever the sinusoid crosses its midline.
 * 3) Sketch U-shapes (facing up at maximums and facing down at the minimums)

__ Steps for Graphing Cotangent: __ >
 * 1) Set the argument (the portion of the equation within the parenthesis) equal to 0 or π to find the first vertical asymptote.
 * 2) Find the period of the graph by using the equation π/b. Remember, b is the coefficient of the variable in the equation. You will always use the absolute value of b.
 * 3) Add and subtract the period from the first vertical asymptote you found, in order to find two additional vertical asymptotes.
 * 4) Sketch two periods of the equation.


 * Formulas:
 * Amplitude: |a|
 * Period: 2π/|b| or π/|b|
 * Frequency: 1/period
 * Phase Shift: -c/|b|
 * Vertical Shift: d
 * Midline: y=d
 * Sine: y=a sin (bx+c)+d
 * Cosine: y=a cos (bx+c)+d

(Alex Travis)