Graphs of rational functions

As a grade 11 student, one of the subjects that you are required to enroll is General Mathematics. In this subject, graphing functions is one of the competencies that is needed to be mastered. One of the functions that will be graphed in this subject is the rational function.

In graphing rational functions manually, it is very important to learn first how to find the intercepts and the asymptotes, and to assign points on each region of the graph. This is not an easy competency to master. However, it is very important for us to learn and understand the basic steps. To review, here are the basic steps in graphing rational functions.

1)      Find the intercepts, if there are any. Remember that the y-intercept is given by (0, y). To find the y-intercept(s), the point(s) where the graph crosses the y-axis, substitute zero for x and solve for y. To find the x-intercept(s), the point(s) where the graph crosses the x-axis, substitute zero for y and solve for x. The x-intercept, also known as the zeros of the function, is given by (x, 0).

2)      Find the vertical asymptotes by setting the denominator equal to zero then solve.

3)      Find the horizontal asymptote, if it exists. Remember that:

·         If both the numerator and denominator are of the same degree, divide the numerical coefficients of the terms with the highest terms in both the numerator and denominator.

·         If the numerator has a degree less than the denominator, the x-axis (y = 0) is the horizontal asymptote.

·         If the numerator has a degree more than the denominator, there is no horizontal asymptote.

4)      Find the vertical asymptote(s) by simply setting the denominator equal to 0 and solve for x. The vertical asymptotes will divide the number line into regions. In each region graph at least one point in each region. This point will tell us whether the graph will be above or below the horizontal asymptote and if we need to we should get several points to determine the general shape of the graph.

5)      Sketch the graph.

Most of the students find this tiresome and mind-numbing. Thus, causing them to fail the subject or even dropping it. To lessen this problem, we can use online graphing calculators and apps. In my class, I prefer to use online graphing calculators which are free and easy to use, even without signing in. We have used Desmos and Meta-calculators in graphing.

For my students, graph 3 rational functions below using the online calculators we have used. Choose one rational function from each set. Be able to give your comment below following the format:

                                                Example:

Name:                                      Juan Dela Cruz

Chosen Rational Function:     f(x) = (3x+2)/(x-1)

URL of the graph:                  https://www.desmos.com/calculator/shevjuhxcy

x – intercept(s):                       (x, 0)

y – intercept(s):                       (0, y)

                  
                 Set A                                       Set B                                                     Set C

A Child's Dream maker

 The dream begins with a teacher who believes in you, who tugs and pushes and leads you to the next plateau, sometimes poking you with a sharp stick called 'truth'. - Dan Rather

            Every child has a dream. This dream can be made or unmade by the people that surround him. One of the people that have the greatest influence on a child's dream is a teacher.

          Since my kindergarten, I have always wanted to become a teacher. They said that a teacher should be competent in all subjects. However, I always fear that I might not be able to make it in Math because I always got the lowest scores in my quizzes. I was sometimes being bullied. Because of that. I opted to be quiet, sit at the back, and hesitate to participate in the class because of the fear that I might give a wrong answer and be bullied again.

          Until, I met my grade 5 teacher in Math. She was very motherly, soft-spoken, yet crack jokes often. We call her Teacher Minda. In her class, whenever we got low scores, she encourages us to try again. After our classes, she asked us to stay. She re teaches her lesson to us making sure that we are able to understand and learn the skill. She does not stop until we are able to get a good score. She told us that Math is fun especially if we keep on practicing it. It was in her class that I started to give my interest in Math. She made me believe that I can do well. And, I did. She brought back my confidence. After grade 5, I was never bullied again.  

I never thought that I would become a Math teacher. When I enrolled in college, I remembered her. She made me believe that I can be what I want to be. I should strive hard to be like her, a Math teacher who made her student’s dream came true.

Synonyms in Quadrilaterals

A. Description:

“Synonyms in a quadrilateral” is an instructional material which enhances students’ knowledge on the principles of diagonals of parallelogram and median of a trapezoid. It is also a means of matching the synonym of the word which can be found on the other half of the quadrilateral. This is an activity which hits two birds in a stone. This is a manner of presenting the lesson that can be both enjoyable and educational on the part of the students. This is done in pairs which helps develop relationships of students in the classroom.  This can be presented after the discussion proper to provide the students follow up on the lesson discussed.

     B. Skills Developed: Vocabulary skills, Spatial Skills

     C. Procedure:

1. The teacher distributes pieces of triangles.

2. Written in each triangle are words that need to be matched with its synonym.

3. The students will look for their partner by matching the triangles to form quadrilaterals.

4. Students will be asked to identify and describe the quadrilateral formed.

5. Each student will also share sentences using the words in their triangles.

6. The teacher will give points to those students who can find their pair in one minute.

Landmarks around the Globe: A Gallery of Solid Figures

  A. Description

Landmarks around the Globe: A Gallery of Solid Figures is an innovative way of presenting a Geometry Lesson in Solid Mensuration.  It is a jigsaw puzzle that presents the different landmarks around the globe which models the different types of solid figures. At the back of each picture, brief information is given, specifically on the dimensions of the landmark. This is a way of presenting the lesson that can be both enjoyable and informational on the part of the students as well as the teacher.  This may also be a means of assessing the students’ spatial intelligence.  The puzzle can be presented before the discussion proper to provide the students time to prepare their minds for the lesson.

  B. Skills Developed: Spatial Skills, Reading

  C. Procedure:    

1. Divide the class into six groups.  Assign each group a leader, a secretary and a reporter.

2. Provide each group with a puzzle pieces, a scotch tape and a Manila paper.

3. Let the students solve the puzzle by fitting each piece correctly to form the picture.

4. Before pasting the solved puzzle in the Manila paper, with the use of the scotch tape, let the students jot down the information about the picture.

5. Aware the students that they will only be given 3 minutes to accomplish the activity.

6. Each group must post their picture on the walls of the room.

7. The reporter will be asked to stand by the wall where the picture is posted to read to the viewers the information about the picture.

8. Each group will be given one minute to roam around the room to see the gallery.

An Improvised Compass

  A. Description:  

An improvised compass is a tool very useful in drawing circles. This will help students some lessons in Geometry and Trigonometry. This will provide students an aid in drawing perfect circles when a compass is not available.  This is an innovation that may be prepared by anyone with materials right at hand.

  B. Materials:

Paper, two ball pens

  C. Skills developed:

Drawing of perfect circles, Resourcefulness

  D. Procedure:

1. Get a ¼ sheet of intermediate paper. Fold it clockwise three times.

2. Get two ball pens. Drive the ball points into the folded paper such that the distance between them corresponds to the desired radius of the circle.

3. With one ball point stationary, move the other ball point until the desired circle is drawn.