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Graphing the rational functions

Course Description:

Examines polynomial and rational functions as well their graphing with analysis of critical properties in the context of real life situations and will include student investigations and hands on activities. 3 credits

Prerequisites:

2 years of high school Algebra
(Students are responsible to review material prerequisite for this course on their own.)
Note: It is highly recommended that "Polynomial Functions" and “Graphing Quadratic Functions” be completed prior to this course with a grade of C or better.

Course Objectives:

a. Polynomial Functions
Use the Fundamental Theorem of Algebra and the Linear Factorization Theorem to write a polynomial as the product of linear factors
Find all real and complex zeros of a polynomial function
Find a polynomial with integer coefficients whose zeros are given
Use the Leading Coefficient Test and the zeros of a polynomial to sketch the graph of a polynomial
Apply techniques for approximating real zeros to solve an application problem

b. Rational Functions
Find the domain of a rational function
Find the vertical and horizontal asymptotes of the graph of a rational function
Sketch the graph of a rational function
Use a rational function model to solve an application problem

The purpose of this course is to increase students' understanding of mathematics, which contributes to a foundation for teaching in K-12. This course emphasizes the concepts and applications of functions, graphical analysis and pre-calculus.

Students should become better problem solvers using the concepts in this course both individually and in working positively with others in solving problems and in the learning process. It is hoped that this course will not only increase the students' knowledge, but also their confidence and enthusiasm to teach K-12 school mathematics.

Required Materials:

Please bring to each class: a copy of the text, the Lab Manual, a graphics calculator, computer 3.5" disk, colored pencils (optional), graph paper 1/4" x 1/4", and a loose leaf notebook. (There is a 5-point penalty for not having materials).

Quizzes and Exams:

There will be 2 exams and a cumulative final exam. Exams are based on the text, supplementary material discussed in class, and on assigned work. Exams will have many problems similar to previous work, but will also contain some novel examples. A request to miss a regularly scheduled exam must be made in advance and must be documented. For a documented excused absence, students may request to have an adjusted average computed instead of taking a make-up exam. Make-up exams will always be more difficult than the regularly scheduled exam. There will be frequent short quizzes on previously discussed material and the quiz average will be equivalent to a third exam.

Homework and Student Presentations:

Students are expected to attempt all of the assigned homework problems in a neat manner, showing all necessary solution steps. Students should seek help prior to class on any homework problems they cannot complete, as they are asked to present homework solutions to the class.

Homework will be spot checked and recorded as "completed assignment or not" during class. Problems with just answers and insufficient solutions steps are considered incomplete. Student presentations and class discussions will center on a few of the homework problems. If additional help is needed, students are advised to come to office hours. Homework should be organized in a loose-leaf notebook and is evaluated again at the end of each unit.

Attendance:

Students are expected to attend all classes. Absences should be only for illness, and documented or recognizable emergencies. The final grade will be lowered one letter grade if a student has missed 10% of classes due to unexcused absences, and students receive an F if they miss more than 25% of the classes for any reason. Students are responsible for all course information during an absence, i.e. securing class notes or handouts, getting extra help, etc. Some of the course materials and labs are not in the text, and some labs require completion at the computer lab outside of class-time.

Journal, Projects, Portfolio & Class Participation:

Students will be asked to write some journals about mathematics. A Portfolio of some of the student's interesting work will be collected near the end of the course. Several special problems and projects will be due as assigned. Much of this course is class discussion of homework or new concepts and working with cooperative partners or groups. Consequently, the "Journal, Homework, Projects, Portfolio & Class Participation" grade is greatly dependent on lively participation with a cooperative and positive attitude. Participation and attitudes, which help the whole class succeed and enjoy mathematics, is highly valued in this course.

Grading:

The grade in this course will be determined using 90%-100% = A, 80%-89% = B, etc. If excused absences would affect a student’s grade, it is the student’s responsibility to present verification of excused absences during office hours near the end of the course. Late work receives half credit if it is received before the next class. The grading scheme and tentative exam schedule are as follows:

Honor Code:

The students in this course should respect and appreciate the Common Honor Code of Learning. They all agreed to assume full responsibility for our actions and will refrain from lying, cheating, stealing and plagiarism and will endeavor to see that others do likewise. While some of the work in this course is cooperative group work, most is individual work. Tutoring and assistance on homework is allowed to help gain understanding; however, the students must actually do the work themselves and be able to explain their process.

Course Information:

Hopefully the students will enjoy this course as much as this instructor does. I think you will find it has a little different format for learning mathematics compared to a traditional math class. Class time will be used for group work and discussions and presentations by myself and by students. Students are expected to be quite active in discussions and in sharing ideas. Students should expect to spend several hours preparing for each class. I welcome you to seek extra help during my office hours; however, I will expect to see serious efforts made by you prior to coming for help. Feel free to stop by. Keeping up to date in this course is extremely important. It has been my experience that students who fail the first exam and who do not have a passing quiz grade will not do well in this course and should seriously consider dropping before the drop date to receive a W grade. Exceptions to the drop date may be made for a major medical emergency and require the deans signature. The syllabus is subject to changes as announced in class. A daily class schedule is available upon request.

“Polynomial and Rational Functions” content delivery

Polynomial Functions

Motivation: [5 minutes] Solving equations is at the heart of any math course. For example, in calculus, to find the critical values students will need to solve equations. Polynomial equations are one example of such equations. Historically, students learned how to find the roots from the given polynomial. Now this can often be done much more easily with the graphing calculator. But the reverse process is also interesting; that is, from the given graph of a polynomial, recover a formula for it. In addition, polynomials are used frequently to model real-life scenarios and make predictions. Polynomial models are often simpler than other models. Solving polynomial equations often leads to simple solutions for real-life applications.

Warm Up Discussion: [5 minute] Provide an example of a polynomial function (in completely factored form) with only real roots. Ask students how many roots it has. Then provide an example of another polynomial function (in completely factored form) with mixed real and complex roots. Ask students how many real roots and how many complex roots it has. Then help students generalize that an nth degree polynomial has n roots.

Find all real and complex zeros of a polynomial function

Warm Up Example or Activity: [20 minutes] Give a polynomial of degree 3 with integer coefficients and one rational root. Use the rational root theorem and synthetic division to find that root. Then use the quadratic formula to find the other complex roots. Point out that complex roots occur in complex conjugate pairs. Point out that this polynomial can be completely factored as a product of three factors. (Then use the graphing calculator to demonstrate that the graph has just one real root; the complex roots do not show as x intercepts.)

Formal Concept: [5 minutes] The Fundamental Theorem of Algebra and the Linear Factorization Theorem can be used to write a polynomial as the product of linear factors. Find a polynomial with integer coefficients whose zeros are given

Warm Up Example or Activity: [20 minutes] Ask students to find a formula for a degree four polynomial with integer coefficients that has two real zeros and one complex zero (a + bi, with b =/ 0). Demonstrate that this polynomial also has the other complex conjugate as a root. Explore different possible solutions, based on the leading coefficient. Have the students graph the functions and observe how changing the sign of the leading coefficient from positive to negative changes the global behavior. (In the previous example, with n = 3, an odd degree, explore how changing the sign of the leading coefficient would change the global behavior of the 3rd degree polynomial.)

Formal Concept: [5 minutes] Have students generalize the Leading Coefficient Test in their own words.

Apply techniques for approximating real zeros to solve an application problem

Example and In-Class Activity: [10 minutes] Have students solve a problem involving a 2nd degree or higher polynomial model for revenue, cost, or profit.

Suggested Follow Up Assessment: the assigned homework problems, and a quiz or warm-up review example at the beginning of the next class.

Rational Functions

Motivation: [5 minutes] Some things grow with limited capacity because of limited space or resources, such as a fish population in a pond. Other things cannot realistically reach 100% optimization, such as pollution removal. Other things decrease over time, such as the concentration of medicine or alcohol in the bloodstream. Rational functions can be used to model these situations and also are used with limits and applications in calculus.

Warm Up Discussion: [5 minutes] One of the most important aspects of rational functions is the concept of vertical and horizontal asymptotes. The graphs of rational functions often are in pieces, with vertical asymptotes (local behavior) at places where the input is not defined and horizontal asymptotes (global behavior). Horizontal asymptotes demonstrate the limiting capacity in applications of rational functions.
Find the domain of a rational function.
Find the vertical and horizontal asymptotes of the graph of a rational function

Warm Up Example or Activity: [20 minutes] Choose a rational function of degree one over degree one in completely simplified form. Ask students for the domain. Remind them that the domain of the function is those real values of x that make the function have meaning. Pick some x values in the domain and the number that is not in the domain. Then talk about the presence of a vertical asymptote on the graph at that x value. Ask students to graph the function to verify this. Demonstrate for them the behavior to the left and right of this value.

Also, ask the students to zoom out to demonstrate the global behavior of the function. Discuss the equation of the horizontal asymptote. Give other quick examples of other cases for horizontal asymptotes, i.e., when the horizontal asymptote is zero or when there is no horizontal asymptote.

Formal Concept: [5 minutes] Have students explain in their own words how to find the domain and the vertical asymptote of a rational function algebraically. Also, lead them to examine and state in their own words how the ratio of the leading terms of the polynomials in the numerator and denominator is related to the equation for the horizontal asymptote.
Sketch the graph of a rational function

Warm Up Examples or Activities: [20 minutes] Give students more examples of higher degree polynomials in the numerator and the denominator to help the students learn to
1. find the y and x intercepts and the domain
2. find the equations of the vertical and horizontal asymptotes
3. select some extra x values to aid in graphing (choose values between vertical asymptotes and the x intercept)
4. graph the function by hand and confirm using your calculator

You may choose an example where the graph intersects the horizontal asymptote locally. (Many students think that the graph cannot intersect the horizontal asymptote.) You may also choose to give an example of a denominator with no real roots and examine the effect this has on the graph.
Use a rational function model to solve an application problem

Warm Up Examples or Activities: [15 minutes] Choose any real applications from the book, e.g., population of animals, pollution removal, drug concentration, average cost, etc. Help students discover that the horizontal asymptote of the function is the limiting capacity (maximum population) or minimum concentration for these kinds of problems.

Suggested Follow Up Assessment: the assigned homework problems, and a quiz or warm-up review example at the beginning of the next class.

Definition of a Rational Function (RF)
A rational function is basically a division (quotient) of two polynomial functions. That is, it is a polynomial divided by another polynomial. In formal notation, a rational function would be symbolized like this:
y = R(x) = g(x) / h(x)
Where g(x) and h(x) are polynomial functions, and h(x) cannot be equal zero.
RFs can have two types of discontinuities: asymptotic discontinuities and hole discontinuities.
Asymptotic discontinuities occur when a denominator h(x) of RF is 0, i.e. h (x a ) = 0.
Hole discontinuities occur when both a numerator g(x) and denominator h(x) are equal to 0,
i.e. g (x h ) = h (x h ) = 0.

The steps of RF analysis:
I. Factorizing and simplifying an original RF.
II. Finding a domain of RF.
III. Finding discontinuities of RF:

1. asymptotic discontinuities
2. hole discontinuities
IV. Intercepts of RF
1. X- Intercepts
2. Y- Intercepts
V. Turning (stationary) points of RF
VI. Sketching the graph of RF
VII. Sign analysis of the graph

Along with polynomial representation the rational functions can be represented in factored form.
Here is an example of a rational function:
y = R(x) = x2-3x / x2-9
Because the graphical examples in the classroom communicate far better than abstractions and generalities, let's address some illustrative rational function f(x) = x2-3x / x2-9 and consider further all the attributes, properties and methods of graphing rational functions following this example.

I. Factorizing and simplifying an original RF.
The very first step is trying to factorize and simplify a given function.
To understand the behavior of a rational function it is very useful to see its polynomials in factored form. Obviously, factorizing f(x)=x(x-3)/(x-3)(x+3) which simplifies to x/x+3. The polynomials in the numerator and the denominator of the above function would factor like this:
R(x) = x(x-3) / (x-3)(x+3) = x / x +3
Take a notice that x=3 gives R(x) = 0 / 0. So, value x=3 provides a pointed, steep type of discontinuity called “hole”. Another type of discontinuity is smooth, non-steep approaching to infinite values called “asymptotes”.

II. Domain
The Domain of RF is the set of all real numbers for which f(x) g(x)/ 0 or f(x) 0/0. In other words, the domain of RF is the intersection of the domains of g(x) and h(x). Now the roots of the denominator are obviously x = -3 and x +3. That is, if x take on either of these two values, the denominator becomes equal to zero. Since one cannot be divided by zero, the function is not defined for these two values of x. We say that the function is discontinuous at x= +3 and x= -3. The domain for the given RF, as expressed in interval notation, is:
D = (- , -3) U (-3, +3) U (+3, + ), where x=-3 and x=+3 are discontinuities.

III. Discontinuities.
Other values for x do not cause the function to become undefined, so, we say that the function is continuous at all other values for x. In other words, all real numbers except -3 and +3 are allowed as inputs to this function. As mentioned above, there are two types of discontinuities: asymptotes and holes.

1. Hole discontinuities
We determined already that hole (steep) discontinuity for the given RF is x=+3 because both g(x) and h(x) have a common factor (x-3) and become equal 0 at x h = 3. If RF is then reduced to lowest terms, the graph of RF has a hole in it where x h = 3. To find the y value, plug x=3 into the simplified function and get 3/6=1/2. The hole is at (3, 1/2). If you have a common zero of g(x) and h(x), this represents a hole in the graph!

2. Asymptotic discontinuities
Asymptotes of a function are lines that the graph of the function gets closer and closer to (but does not actually touch), as one travels out along that line in either direction. Generally, there are three types of asymptotes: vertical, horizontal and oblique (slant).

a. Vertical asymptotes
The vertical asymptotes for a RF are determined by the zeros of the denominator (i.e. the values for which the denominator equals 0). Find the zeros of the denominator after you simplified. You can find the vertical asymptotes by equating the denominator to 0 and solving, and then see if y approaches infinity or negative infinity on each side of the potential asymptote.
Find the zeros of h (x). These will be the vertical asymptotes unless it's also a zero of g (x)!
Set h (x) = 0 and solve for x.
A vertical asymptote for RF is a vertical line x=k; k. is a constant, that the graph of RF approaches but does not touch. For the given RF the vertical asymptote is x v a = - 3.

b. Horizontal asymptotes
A RF has a horizontal asymptote y = a, if; as |x| increases without limit y approaches a. RF y = f(x) has at most one horizontal asymptote. The horizontal asymptote may be found from a comparison of the degree of g(x) and the degree of h(x).
Find the horizontal asymptotes of the function after you simplify.
a. if degree(h(x)) > degree(g(x)) then horizontal asymptote y =0;
b. if degree(h(x)) = degree(g(x)) then horizontal asymptote y = a/b (leading coefficient of g(x))/ (leading coefficient of h(x));
c. if degree(h(x)) < degree(g(x)) then no horizontal asymptote

The graph of y=f(x) may cross a horizontal asymptote in the interior of its domain. This is possible since we are only concerned with how RF behaves as |x| increases without limit in determining the horizontal asymptote.

Lim x/x+3 = Lim (x / x) / (1+1/x) = 1/1=1
x x
The horizontal asymptote is y h a = 1.
The horizontal asymptotes of a function can be found by dividing both the numerator and denominator of the rational function by the highest power of x that appears in the denominator. You will then likely produce at least one term of the form c/x n. As x approaches infinity (positive or negative), this term approaches zero, thus it can be eliminated from the expression, and you can solve for y to find the horizontal asymptotes.

C. Slant (oblique) asymptotes
Utilize polynomial algebraic division. In this case: R (x) = x / x + 3.
Linear oblique asymptote like R(x) = kx + a will occur if degree (h(x)) = degree (g(x)) – 1.
Because this condition fails there is no oblique asymptotes for the given rational function.

IV. Intercepts
1. X-Intercepts
The x-intercepts (if any) of y are the zeros of the numerator, p(x), since the function is zero only when its numerator is 0. R(x) = g(x) / h(x); if R(x) = 0 then g(x) = 0
Find the zeros of g(x). These will be the x-intercepts unless it's also a zero of h(x)!
Find the x-intercept by finding the zeros of the numerator: x int = 0.

2. Y=Intercepts
Find the y-intercept by replacing the x value with 0. R(x=0) = g(x=0) / h(x=0) = 0 / 3 = 0.
If the denominator is not zero, you have found the y-intercept - y int = 0!

V. Turning (stationary) points
Turning (stationary) points - extremes (local relative minimum or maximum)
To find extrema you have to equal a function derivative to 0 and determine X-coordinate of a function extreme. With usage of a known formula for a derivative (division of two polynomials Y=U/V):
Y’ = (VU’ - UV') / V2
For the referred example:
Y = (x2 - 3x) / (x2 - 9)
Y' = 1 / 3(x+3)2
There are no extrema because Y’ 0 (never equal to 0)

VI. Sketching the graph
Graph sketch: Use the vertical and horizontal asymptotes to help sketch the graph. If x = c is a vertical asymptote, then the graph approaches infinity as it nears the asymptote in a region where the function is positive, and it approaches negative infinity as it nears the asymptote in a region where the function is negative. (Note: vertical asymptotes cannot be crossed because they describe where the graph is undefined. Horizontal asymptotes may be crossed as they describe only what happens to the graph as x gets very large or very small!)
a) Find the zeros of the denominator after you simplified. The vertical asymptote is x = -3.
b) Find the limits on infinity of the function after you simplify.

The horizontal asymptote is y = 1.
c) Find the x-intercept by finding the zeros of the numerator. x =0
d) Find the y-intercept by replacing the x value with 0. y-int is 0.
e) Find the holes in the graph. The hole appears at the factor that was canceled. In this case, at x =3.
To find the y value, plug x=3 into the simplified function and get 3/6 = 1/2. The hole is at (3, 1/2).

VII. Sign analysis
Do a sign analysis in each interval separated by asymptotes and intercepts. Sign analysis for each area of the graph. In the upper left corner, the y values are + which means the graph is above the x-axis approaching the vertical and horizontal asymptote. The area to the left of the zero is negative and approaches the vertical axis downward. The area to the right of the zero reverts back to positive and approaches the horizontal asymptote. It will not cross the horizontal asymptote because setting 1 = x/(x+3) yields no solutions! The hole y h = 0/0 not allowed to divide 0 by 0 (infinity).

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Alexander C.

Mathematics Tutor