Rational+Functions

** Unit 3: Rational Functions **



 * Graphing and Characteristics **


 * __ Unit 3A Summary __ (Alex T.) **

Unit 3A is devoted to determining the characteristics and graphing rational functions without the use of a calculator. The pre-requisite skill required for this unit is a knowledge of factoring, whether by simply factoring, factoring by grouping or using the AC method.

Factoring: Ex. x^2-6x-16 can be factored to (x+2) (x-8) In order to factor this, you must think of factors of -16 that would have a sum of -6. This is simple because there is no leading coefficient. To factor by grouping, you must group terms with common factors together, then factor out their greatest common factor. With this done, there should be the same value left in both parenthesis, one of these parenthesis is disregarded, and the two values left outside the parenthesis are inserted into their own set of parenthesis. These two binomials in parenthesis are your factored product. You start the AC method by multiplying the "a", the squared term, and the "c" term, the term lacking a variable. Then, you must find the factors of the product of "a" and "c" that would result in the sum of the "b" term, the term with the "x" variable. Once these two factors have been established, insert them into the trinomial in place of the "b" value. Lastly, simply factor by grouping. Factoring is extremely important when it comes to determining the characteristics of the graph.
 * Factoring by Grouping: Ex. x^3+2x^2+8x+16= (x^3+2x^2) (8x+16)= x^2(x+2) 8(x+2)= (x^2+8) (x+2)
 * AC Method: Ex. 2x^2+7x-15= 2x^2+10x-3x-15= 2x(x+5) -3(x+5)= (2x-3) (x+5)

In Unit 3A we discovered how to find various characteristics of the graphs of rational functions such as vertical asymptotes, horizontal asymptotes, holes, x-intercepts, y-intercepts, domain, range, and end behavior. lim (insert function name)= (insert the value of the horizontal asymptote, or if it has a slant asymptote, positive or negative infinity) x approaches (insert a positive infinity then a negative infinity, or if the end behavior is the same, insert a positive and negative infinity)
 * Vertical asymptotes: once the denominator of the equation is completely factored using one of the methods above, set it equal to zero and solve for x. This x value should be placed on the graph as a dashed line that passes through the x-axis at that value.
 * Horizontal asymptotes: The horizontal asymptote depends on the degrees of the terms in the rational function. There are three rules:
 * 1) Top heavy: The highest degree of the function is in the numerator, so use long division, dividing the numerator by the denominator, in order to find the formula of the line that will be the slant asymptote. This is represented by a dashed line on the graph.
 * 2) Bottom heavy: The highest degree of the function is in the denominator, so the horizontal asymptote is y=0, and is also represented by a dashed line that happens to be the x-axis.
 * 3) Same degree: The highest degree can be found on both the numerator and the denominator, so take the leading coefficients of the variables with the same degree and create a fraction with the numerator's term on top and the denominators term on the bottom. This fraction is where you would draw your horizontal asymptote that is also represented by a dashed line and passes through the y-axis.
 * Holes: Holes are found by completely factoring the numerator and denominator of the rational function, then cancelling out any binomial that appear in both the numerator and denominator. This cancelling will ultimately affect the other characteristics of the graph, so this must be done before finding anything other characteristics. To find the actual point of the hole, you must set the cancelled binomial equal to zero and solve for x. Upon finding that term, you would plug it in for x in the remainder of the function that has not been cancelled, the value this produces is the hole's y-coordinate. Therefore, the hole's x-coordinate is what was found by setting the cancelled portion equal to zero, and the y-coordinate is the result of plugging the x-coordinate into the remainder of the function as x and the solving. Holes are plotted as unfilled in circles.
 * X-intercepts: The x-intercept(s) are found by setting the factored numerator of the rational function equal to zero, then solving for x. There may be multiple answers, and each is plotted as a solid dot on the x-axis. The coordinates of the x-intercept are determined by the value found being the x-coordinate, and zero being the y-coordinate.
 * Y-intercepts: To find the y-intercept of a rational function, substitute all of the x's in the function for zeros, then solve. This point is the y-coordinate of the point, and the x-coordinate is zero. it is represented as a solid point on the y-axis.
 * Domain: Domain is found by listing all x values possible from left to right, although, you must jump over holes and vertical asymptotes. This results in union between two coordinate pairs being necessary in the domain. You must union the previous x values with those after the interruption in the form of a vertical asymptote or hole. Domain is written in interval notation with parenthesis when a value is excluded, and brackets when a value is included. Most graphs include infinity and negative infinity in the domain.
 * Range: Range is very similar to domain. You must list all possible y values from top to bottom, while avoiding holes and horizontal asymptotes in the same way domain jumps over holes and vertical asymptotes. Range is also written in interval notation with brackets on included values and parenthesis on excluded values, and most ranges include positive and negative infinity.
 * End Behavior: End Behavior describes the ends of each branch of the graph. As x moves towards positive or negative infinity, the y values moves towards and follows the horizontal asymptote or slant asymptote. Domain, range, and end behavior are found after graphing the rational function. How to write the end behavior:
 * Steps to Graph Rational Functions:
 * 1) Graph the vertical and horizontal asymptotes
 * 2) Plot the x and y intercepts
 * 3) Sketch to avoid more x-intercepts
 * 4) If more information is needed, pick points to plot to the left, right, and between the vertical asymptotes.



= = A rational function is defined as the quotient of two polynomial functions and is written as a fraction or //ratio//, hence the name. They can be described as "top--heavy", "bottom-heavy", or "same degree". In the case of top-heavy rational functions, the degree or highest exponential factor of the numerator is higher than that of the denominator. In bottom heavy functions, the degree of the denominator is higher. And as the name suggests, the denominator and numerator of same degree rational functions are equivalent.
 * __ Summary of Rational Functions __**** (Kelsey P.) **

To graph a rational function, you must first find out what its asymptotes are. An asymptote is an invisible line on the coordinate plane that the branches of the graph will approach but never touch. Asymptotes can be vertical, horizontal, or oblique (slant). Then you must look for any intercepts, or points where the graph crosses the x or y axis. Next you may want to plot a few extra points to be sure of what shape the graph will be taking. Finally, you can sketch in the graph of the function.

__** Rules for finding horizontal asymptotes: **__
 * For bottom heavy functions, it is always y=0.




 * For same degree functions, you must take the ratio of the coefficients.

**x=2**




 * For top heavy functions, there will be a slant asymptote. To find this, you must use long division.

** y=x-5 **



__** Step-By-Step on How To Graph a Rational Function: **__

1.) To find any vertical asymptotes, factor if possible and set the denominator equal to zero.

x-1=0, x=1

2.) To find possible horizontal asymptotes of a same degree function, you must take the ratio of the leading coefficients.

2/1, which can simplify to y=2



3.) Now we need to find any x or y intercepts. To find the x intercept, set the numerator equal to zero. 2x+5=0 2x=-5 x=-5/2 so, the x intercept can be found at (-5/2, 0)   To find the y intercept, set x equal to zero.  (2(0)+5)/(0-1) y=-5 so, the y intercept can be found at (0,-5)

4.) Now we can sketch in the final graph:


 * __ Instructional Videos __**** (Daniel R.) **


 * [] This video shows you step-by-step how to graph a "same degree" rational function. Besides showing you clear visuals and representations of what the final graph looks like, it also teaches you the steps used to find the many characteristics of the graph/function. Some of the characteristics include the horizontal and vertical asymptotes and the x and y intercepts. This specific does not contain any holes and is divided into three sections. (Daniel Reyes)[[image:seigleprecalculus/same degree.jpg]](Daniel Reyes)


 * [] If you have any problems in understanding the three basic rules of how to find the horizontal asymptotes, then this video is suitable for you. The presenter goes into great detail explaining the difference between each rule and demonstrates different hints that will definitely help you graph a horizontal asymptote in an instant. Furthermore, she also demonstrates why each rule works and gives you examples of each rule in an easy-way to understand. (Daniel Reyes)


 * [] This video does a great job of explaining what you need to do in order to find the vertical asymptotes of rational functions. It explains how it is necessary to set the denominator equal to zero after factoring as much as possible. One important thing to note is that in this video, the person uses a different method of deciding whether a vertical asymptote is real or not. He uses a zero over zero idea to show an unreal asymptote and a "number" over zero idea to show a real asymptote. In class, we factored first the numerator and the denominator to cancel out any holes present. By doing this, we skipped that last step showed in the video. (Daniel Reyes)

(Daniel Reyes)
 * [] This video is simply excellent. It starts off with a brief background explanation of what the end behavior is and how it relates to a rational function. Then, the video not only continues to explain end behavior, but it also talks about the x and y intercepts and the vertical and horizontal asymptotes. The only thing that the video lacks is that it doesn't mention the end behavior in limit notation, but other than that, this video can even serve as an overall review of rational functions. (Daniel Reyes)


 * [] Here is video that explains how to graph rational functions that are "top-heavy", in which the degree of the numerator is larger than that of the denominator. When this is the case, the graph would have not a horizontal asymptote, but a slant asymptote of y=mx+b. You can solve for the slant asymptote using two methods, synthetic division or long division as shown in the video. (Kelsey Parkhill)


 * [] This video shows you how to find a possible hole in the graph. In other words, this is a point that the graph will "jump over" as is approaches positive or negative infinity. The video explains step by step on how to find a hole, from factoring the polynomials to finding common factors between the numerator and the denominator. (Kelsey Parkhill)


 * __ Helpful Sites __ (Kayla B.) **

@http://www.purplemath.com/modules/grphrtnl.htm

This website proved to be very helpful in teaching the basic introduction to graphing rational functions. It shows you, step-by-step, how to find your asymptotes, intercepts, and overall shape of the graph. This website includes many examples, graphs, and procedures that relate to the process of graphing rational functions.

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On this website, the graphing of rational functions is explained in full detail. Not only are examples given, but definitions and concepts are also explained. The examples given on this website range from being top heavy, bottom heavy, and same degree in order to fully explain the process.

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Looking through this website, you will be able to strengthen your ability to simplify rational functions. Multiple methods are explained in order to provide alternative ways to explain/solve rational equations. It also provides a few problems that act as a self-evaluation after reading the given information.

[|http://www.edurite.com/kbase/how+to+solve+an+equation+with+rational+expressions#]

This pre-calculus website offers a plethora of helpful tools concerning and understanding rational equations. It contains several sections that each go into full detail describing every process necessary. These sections, notes, and demos provide great examples.

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If you prefer to see examples and answers when understanding a math topic, this site would be helpful to you. As mentioned, several examples are given and then broken down into multiple questions. Following these questions, the answers to each section are given with detailed explanations.




 * Key for Practice Problems: [[file:Unit 3 Wiki Group - Practice Problems - KEY by Alex King.pdf]] **


 * __ Jobs and Occupations that use Rational Functions __**** (Emma W.) **


 * Envrionmental Engineer**

Design, plan, or perform engineering duties in the prevention, control, and remediation of environmental health hazards, using various engineering disciplines.Work may include waste treatment, site remediation, or pollution control technology. For example, an Environmental Engineer may work for a utility company that burns coal to generate electricity. The utility may use a cost-benefit model for removing pollutants from smokehouse emissions. This cost-benefit model is represented by the following equation:

C = 50,000//p// / (100 - //p// ) where C is the cost (in dollars) and //p// is the % of smokehouse pollutants

If the current law requires the utility to remove 85% of the pollutants and is considering changing the law to require 90% of the pollutants to be removed, an Environmental Engineer could use the above cost-benefit equation to determine the cost impact on the utility of the law change.

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 * Biological Scientist**

Biological scientists study living things and their environment. They do research, develop new medicines, increase crop amounts, and improve the environment. Some study specialty areas such as viruses, ocean life, plant life, or animal life. They may work in company, college, or government labs, or as high school biology teachers. For example, a biological scientist may work for the game commission for a state government. This particular state may want to introduce deer to an area of the state with newly acquired state lands. The population of the deer herd (N) could be modeled by the following equation:

N = 20(5 + 3//t//) / (1 + 0.04//t//)where //t// is the time in years

The biological scientist could use the above rational function to predict when in the future the game commission could open the land for hunting.

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Beardon, Alan F. Graduate Texts in Mathematics. San Francisco: Springer-Verlag, 1916. Text.
 * __ Annotated Bibliography __**** (Jason C.) **