“SUBASH BASIC EDUCATIONAL SCHOOL” BALTASI MUNICIPAL DISTRICT
REPUBLIC OF TATARSTAN
Lesson Development - Grade 9
Topic: Fractional linear function
GarifullinaRailIRifkatovna
201 4
Lesson topic: Fractional - linear function.
The purpose of the lesson:
Educational: Introduce students to the conceptsfractional - linear function and equation of asymptotes;
Developing: Formation of techniques logical thinking, the development of interest in the subject; develop finding the area of definition, the area of value fractionally - linear function and the formation of skills for building its schedule;
- motivational goal:education of mathematical culture of students, mindfulness, preservation and development of interest in the study of the subject through the application various forms mastery of knowledge.
Equipment and literature: Laptop, projector, interactive whiteboard, coordinate plane and graph of the function y= , reflection map, multimedia presentation,Algebra: textbook for grade 9 basic secondary school/ Yu.N. Makarychev, N.G. Mendyuk, K.I. Neshkov, S.B. Suvorova; under the editorship of S.A. Telyakovsky / M: “Enlightenment”, 2004 with additions.
Lesson type:
lesson on improving knowledge, skills, skills.
During the classes.
Target: - development of oral computing skills;
repetition of theoretical materials and definitions necessary for the study of a new topic.
Good afternoon! We start the lesson by checking homework:
Attention to the screen (slide 1-4):
Exercise 1.
Please answer question 3 according to the graph of this function (find highest value functions, ...)
( 24 )
Task -2. Calculate the value of the expression:
-
=
Task -3: Find triple the sum of the roots quadratic equation:
X 2 -671∙X + 670= 0.
The sum of the coefficients of the quadratic equation is zero:
1+(-671)+670 = 0. So x 1 =1 and x 2 = Consequently,
3∙(x 1 +x 2 )=3∙671=2013
And now we will write sequentially the answers to all 3 tasks through dots. (24.12.2013.)
Result: Yes, that's right! And so, the topic of today's lesson:
Fractional - linear function.
Before driving on the road, the driver must know the rules traffic: prohibition and permission signs. Today we also need to remember some prohibiting and allowing signs. Attention to the screen! (Slide-6
)
Conclusion:
The expression doesn't make sense;
Correct expression, answer: -2;
correct expression, answer: -0;
you can't divide by zero 0!
Pay attention to whether everything is written correctly? (slide - 7)
1) ; 2) = ; 3) = a .
(1) true equality, 2) = - ; 3) = - a )
II. Exploring a new topic: (slide - 8).
Target: To teach the skills of finding the area of definition and the area of \u200b\u200bvalue of a fractional-linear function, plotting its graph using parallel transfer of the graph of the function along the abscissa and ordinates.
Determine which function is graphed on coordinate plane?
The graph of the function on the coordinate plane is given.
Question
Expected response
Find the domain of the function, (D( y)=?)
X ≠0, or(-∞;0]UUU
We move the graph of the function using parallel translation along the Ox axis (abscissa) by 1 unit to the right;
What function is graphed?
We move the graph of the function using parallel translation along the Oy (ordinate) axis by 2 units up;
And now, what function graph was built?
Draw lines x=1 and y=2
What do you think? What direct lines did we get?
It's those straight lines, to which the points of the curve of the graph of the function approach as they move away to infinity.
And they are calledare asymptotes.
That is, one asymptote of the hyperbola runs parallel to the y-axis at a distance of 2 units to its right, and the second asymptote runs parallel to the x-axis at a distance of 1 unit above it.
Well done! Now let's conclude:
The graph of a linear-fractional function is a hyperbola, which can be obtained from the hyperbola y =by using parallel transfers along the coordinate axes. To do this, the formula of a linear-fractional function must be represented in following form: y=
where n is the number of units by which the hyperbola moves to the right or left, m is the number of units by which the hyperbola moves up or down. In this case, the asymptotes of the hyperbola are shifted to the lines x = m, y = n.
Here are examples of a fractional linear function:
; .
Fractional linear function is a function of the form y = , where x is a variable, a, b, c, d are some numbers, with c ≠ 0, ad - bc ≠ 0.
c≠0 andad- bc≠0, since at c=0 the function turns into a linear function.
If aad- bc=0, we get a reduced fraction value, which is equal to (i.e. constant).
Properties of a linear-fractional function:
1. When increasing positive values argument, the values of the function decrease and tend to zero, but remain positive.
2. As the positive values of the function increase, the values of the argument decrease and tend to zero, but remain positive.
III - consolidation of the material covered.
Target: - develop presentation skills and abilitiesformulas of a linear-fractional function to the form:
To consolidate the skills of compiling asymptote equations and plotting a fractional linear function.
Example -1:
Solution: Using transforms this function represent in the form .
=
(slide-10)
Physical education:
(warm-up leads - duty officer)
Target: - Removing mental stress and strengthening the health of students.
Work with the textbook: No. 184.
Solution: Using transformations, we represent this function as y=k/(х-m)+n .
=
de x≠0.
Let's write the asymptote equation: x=2 and y=3.
So the graph of the function moves along the x-axis at a distance of 2 units to its right and along the y-axis at a distance of 3 units above it.
Group work:
Target: - the formation of skills to listen to others and at the same time specifically express their opinion;
education of a person capable of leadership;
education in students of the culture of mathematical speech.
Option number 1
Given a function:
.
.
Option number 2
Given a function
1. Bring the linear-fractional function to the standard form and write down the asymptote equation.
2. Find the scope of the function
3. Find the set of function values
1. Bring the linear-fractional function to the standard form and write down the asymptote equation.
2. Find the scope of the function.
3. Find a set of function values.
(The group that completed the work first is preparing to defend group work at the blackboard. An analysis of the work is being carried out.)
IV. Summing up the lesson.
Target: - analysis of theoretical and practical activities on the lesson;
Formation of self-esteem skills in students;
Reflection, self-assessment of activity and consciousness of students.
And so, my dear students! The lesson is coming to an end. You have to fill out a reflection map. Write your opinions clearly and legibly
Last name and first name ________________________________________
Lesson stages
Determination of the level of complexity of the stages of the lesson
Your us-triple
Evaluation of your activity in the lesson, 1-5 points
light
medium heavy
difficult
Organizational stage
Learning new material
Formation of skills of the ability to build a graph of a fractional-linear function
Group work
General opinion about the lesson
Target: - verification of the level of development of this topic.
[p.10*, No. 180(a), 181(b).]
Preparation for the GIA: (Working on “Virtual elective” )
Exercise from the GIA series (No. 23 - maximum score):
Plot the function Y=
and determine for what values of c the line y=c has exactly one common point with the graph.
Questions and tasks will be published from 14.00 to 14.30.
Fractional rational function
Formula y = k/x, the graph is a hyperbola. In Part 1 of the GIA, this function is proposed without offsets along the axes. Therefore, it has only one parameter k. The biggest difference in appearance graphics depend on the sign k.
It's harder to see the differences in the graphs if k one character:
As we can see, the more k, the higher the hyperbole goes.
The figure shows functions for which the parameter k differs significantly. If the difference is not so great, then it is quite difficult to determine it by eye.
In this regard, the following task, which I found in a generally good guide for preparing for the GIA, is simply a “masterpiece”:
Not only that, in a rather small picture, closely spaced graphs simply merge. Also, hyperbolas with positive and negative k are depicted in the same coordinate plane. Which is completely disorienting to anyone who looks at this drawing. Just a "cool star" catches the eye.
Thank God it's just a training task. In real versions, more correct wording and obvious drawings were offered.
Let's figure out how to determine the coefficient k according to the graph of the function.
From the formula: y = k / x follows that k = y x. That is, we can take any integer point with convenient coordinates and multiply them - we get k.
k= 1 (- 3) = - 3.
Hence the formula for this function is: y = - 3/x.
It is interesting to consider the situation with fractional k. In this case, the formula can be written in several ways. This should not be misleading.
For example,
It is impossible to find a single integer point on this graph. Therefore, the value k can be determined very roughly.
k= 1 0.7≈0.7. However, it can be understood that 0< k< 1. Если среди предложенных вариантов есть такое значение, то можно считать, что оно и является ответом.
So let's summarize.
k> 0 the hyperbola is located in the 1st and 3rd coordinate angles (quadrants),
k < 0 - во 2-м и 4-ом.
If a k modulo greater than 1 ( k= 2 or k= - 2), then the graph is located above 1 (below - 1) on the y-axis, looks wider.
If a k modulo less than 1 ( k= 1/2 or k= - 1/2), then the graph is located below 1 (above - 1) along the y-axis and looks narrower, “pressed” to zero:
Consider the questions of the methodology for studying such a topic as "plotting a graph of a fractional linear function." Unfortunately, its study has been removed from basic program and a math tutor in his classes does not touch it as often as he would like. However, no one has yet canceled mathematical classes, the second part of the GIA too. Yes, and in the Unified State Examination, there is a possibility of its penetration into the body of the C5 task (through the parameters). Therefore, you will have to roll up your sleeves and work on the method of explaining it in a lesson with an average or moderately strong student. As a rule, a math tutor develops explanation techniques for the main sections school curriculum during the first 5-7 years of operation. During this time, dozens of students of various categories manage to pass through the eyes and hands of the tutor. From neglected and naturally weak children, loafers and truants to purposeful talents.
Over time, the mastery of explanation comes to a math tutor complex concepts plain language without compromising mathematical completeness and accuracy. An individual style of presentation of material, speech, visual accompaniment and registration of records is developed. Any experienced tutor will tell the lesson with his eyes closed, because he knows in advance what problems arise with understanding the material and what is needed to resolve them. It is important to choose Right words and notes, examples for the beginning of the lesson, for the middle and end, as well as correctly compose exercises for homework.
Some particular methods of working with the topic will be discussed in this article.
What graphs does a math tutor start with?
You need to start with a definition of the concept under study. I remind you that a fractional linear function is a function of the form . Its construction is reduced to the construction the most common hyperbole by well-known simple techniques for converting graphs. 
In practice, they are simple only for the tutor himself. Even if a strong student comes to the teacher, with a sufficient speed of calculations and transformations, he still has to tell these techniques separately. Why? At school, in the 9th grade, graphs are built only by shifting and do not use methods for adding numerical factors (compression and stretching methods). What chart is used by the math tutor? 
What is the best place to start? All preparation is carried out on the example of the most convenient, in my opinion, function 
. What else to use? Trigonometry in the 9th grade is studied without graphs (and they do not pass at all in the converted textbooks under the conditions of the GIA in mathematics). quadratic function does not have in this topic the same "methodological weight" that has a root. Why? In the 9th grade, the square trinomial is studied thoroughly and the student is quite capable of solving construction problems without shifts. The form instantly causes a reflex to open the brackets, after which you can apply the rule of standard plotting through the top of the parabola and the table of values. With such a maneuver it will not be possible to perform and it will be easier for the math tutor to motivate the student to study the general methods of transformation. Using the y=|x| also does not justify itself, because it is not studied as closely as the root and schoolchildren are terribly afraid of it. 
In addition, the module itself (more precisely, its "hanging") is among the studied transformations.
So, the tutor is left with nothing more convenient and effective than to prepare for the transformations with the help of square root. It takes practice to build graphs like this. Let us assume that this preparation was a success. The child knows how to shift and even compress / stretch charts. What's next?
The next stage is learning to select the whole part. Perhaps this is the main task of a math tutor, because after the whole part is highlighted, she takes on the lion's share of the entire computational load on the topic. It is extremely important to prepare a function for a form that fits into one of the standard construction schemes. It is also important to describe the logic of transformations in an accessible, understandable way, and on the other hand, mathematically accurate and harmonious.
Let me remind you that in order to plot a graph, you need to convert a fraction to the form 
. To this, and not to 
, keeping the denominator. Why? It is difficult to perform transformations of the graph, which not only consists of pieces, but also has asymptotes. Continuity is used to connect two or three more or less clearly moved points with one line. In the case of a discontinuous function, it is not immediately clear which points to connect. Therefore, compressing or stretching a hyperbole is extremely inconvenient. A math tutor is simply obliged to teach a student to manage with shifts alone.
To do this, in addition to highlighting the integer part, you also need to remove the coefficient in the denominator c.
Extracting the integer part of a fraction
How to teach the selection of the whole part? Mathematics tutors do not always adequately assess the level of knowledge of a student and, despite the absence of a detailed study of the theorem on dividing polynomials with a remainder in the program, they apply the rule of dividing by a corner. If the teacher takes up the corner division, then you will have to spend almost half of the lesson explaining it (unless, of course, everything is carefully substantiated). Unfortunately, the tutor does not always have this time available. Better not to think about any corners at all.
There are two ways to work with a student:
1) The tutor shows him the finished algorithm using some example of a fractional function.
2) The teacher creates conditions for the logical search for this algorithm.
The implementation of the second way seems to me the most interesting for tutoring practice and extremely useful to develop the student's thinking. With the help of certain hints and indications, it is often possible to lead to the discovery of a certain sequence of correct steps. In contrast to the automatic execution of a plan drawn up by someone, a 9th grade student learns to look for it on his own. Naturally, all explanations must be carried out with examples. Let's take a function for this and consider the tutor's comments on the algorithm's search logic. A math tutor asks: “What prevents us from performing a standard graph transformation by shifting along the axes? Of course, the simultaneous presence of X in both the numerator and the denominator. So you need to remove it from the numerator. How to do this with identical transformations? There is only one way - to reduce the fraction. But we don't have equal factors (brackets). So you need to try to create them artificially. But how? You cannot replace the numerator with the denominator without any identical transition. Let's try to convert the numerator so that it includes a bracket equal to the denominator. Let's put it there forcibly and “overlay” the coefficients so that when they “act” on the bracket, that is, when it is opened and similar terms are added, a linear polynomial 2x + 3 would be obtained.
The math tutor inserts gaps for the coefficients in the form of empty rectangles (as is often used in textbooks for grades 5-6) and sets the task of filling them in with numbers. The selection should be from left to right starting from the first pass. The student must imagine how he will open the bracket. Since its disclosure will result in only one term with x, then it is its coefficient that should be equal to the highest coefficient in the old numerator 2x + 3. Therefore, it is obvious that the first square contains the number 2. It is filled. A math tutor should take a fairly simple fractional linear function with c=1. Only after that you can proceed to the analysis of examples with an unpleasant form of the numerator and denominator (including those with fractional coefficients).
Move on. The teacher opens the bracket and signs the result right above it. 
You can shade the corresponding pair of factors. To the "expanded term", it is necessary to add such a number from the second gap to get the free coefficient of the old numerator. Obviously it's 7.
Next, the fraction is broken down into the sum of individual fractions (usually I circle the fractions with a cloud, comparing their location with butterfly wings). And I say: "Let's break the fraction with a butterfly." Students remember this phrase well. 
The math tutor shows the whole process of extracting the integer part to the form to which it is already possible to apply the hyperbola shift algorithm: 
If the denominator has a senior coefficient that is not equal to one, then in no case should it be left there. This will bring both the tutor and the student an extra headache, associated with the need for an additional transformation, and the most difficult one: compression - stretching. For the schematic construction of a graph of direct proportionality, the type of numerator is not important. The main thing is to know his sign. Then it is better to transfer the highest coefficient of the denominator to it. For example, if we are working with the function 
, then we simply take 3 out of the bracket and “raise” it into the numerator, constructing a fraction in it. We get a much more convenient expression for construction: It remains to shift to the right and 2 up.
If a “minus” appears between the integer part 2 and the remaining fraction, it is also better to put it in the numerator. Otherwise, at a certain stage of construction, you will have to additionally display the hyperbola relative to the Oy axis. This will only complicate the process.
Math Tutor's Golden Rule:
all inconvenient coefficients leading to symmetries, contractions or expansions of the graph must be transferred to the numerator.
It is difficult to describe the techniques of working with any topic. There is always a feeling of some understatement. How much you managed to talk about a fractional linear function is up to you to judge. Send your comments and feedback to the article (you can write them in the box that you see at the bottom of the page). I will definitely publish them.
Kolpakov A.N. Mathematics tutor Moscow. Strogino. Methods for tutors.
In this lesson, we will consider a linear-fractional function, solve problems using a linear-fractional function, module, parameter.
Theme: Repetition
Lesson: Linear Fractional Function
Definition:
A linear-fractional function is called a function of the form:
For example:
Let us prove that the graph of this linear-fractional function is a hyperbola.
Let's take out the deuce in the numerator, we get:
We have x in both the numerator and the denominator. Now we transform so that the expression appears in the numerator:
Now let's reduce the fraction term by term:
Obviously, the graph of this function is a hyperbola.
We can offer a second way of proof, namely, divide the numerator by the denominator into a column:
Got:
It is important to be able to easily build a graph of a linear-fractional function, in particular, to find the center of symmetry of a hyperbola. Let's solve the problem.
Example 1 - sketch a function graph:
We have already converted this function and got:
To build this graph, we will not shift the axes or the hyperbola itself. We use the standard method of constructing function graphs, using the presence of intervals of constancy.
We act according to the algorithm. First, we examine the given function.
Thus, we have three intervals of constancy: on the far right () the function has a plus sign, then the signs alternate, since all roots have the first degree. So, on the interval the function is negative, on the interval the function is positive.
We build a sketch of the graph in the vicinity of the roots and break points of the ODZ. We have: since at the point the sign of the function changes from plus to minus, then the curve is first above the axis, then passes through zero and then is located under the x-axis. When the denominator of a fraction is practically zero, then when the value of the argument tends to three, the value of the fraction tends to infinity. AT this case, when the argument approaches the triple on the left, the function is negative and tends to minus infinity, on the right, the function is positive and exits from plus infinity.
Now we build a sketch of the graph of the function in the vicinity of infinitely distant points, i.e. when the argument tends to plus or minus infinity. In this case, the constant terms can be neglected. We have:
Thus, we have a horizontal asymptote and a vertical one, the center of the hyperbola is the point (3;2). Let's illustrate:
Rice. 1. Graph of a hyperbola for example 1
Problems with a linear-fractional function can be complicated by the presence of a module or parameter. To build, for example, a function graph, you must follow the following algorithm:
Rice. 2. Illustration for the algorithm
The resulting graph has branches that are above the x-axis and below the x-axis.
1. Apply the specified module. In this case, the parts of the graph that are above the x-axis remain unchanged, and those that are below the axis are mirrored relative to the x-axis. We get:
Rice. 3. Illustration for the algorithm
Example 2 - plot a function graph:
Rice. 4. Function graph for example 2
Let's consider the following task - to plot a function graph. To do this, you must follow the following algorithm:
1. Graph the submodular function
Suppose we have the following graph:
Rice. 5. Illustration for the algorithm
1. Apply the specified module. To understand how to do this, let's expand the module.
Thus, for function values with non-negative values of the argument, there will be no changes. Regarding the second equation, we know that it is obtained by a symmetrical mapping about the y-axis. we have a graph of the function:
Rice. 6. Illustration for the algorithm
Example 3 - plot a function graph:
According to the algorithm, first you need to plot a submodular function graph, we have already built it (see Figure 1)
Rice. 7. Function graph for example 3
Example 4 - find the number of roots of an equation with a parameter:
Recall that solving an equation with a parameter means iterating over all the values of the parameter and specifying the answer for each of them. We act according to the methodology. First, we build a graph of the function, we have already done this in the previous example (see Figure 7). Next, you need to cut the graph with a family of lines for different a, find the intersection points and write out the answer.
Looking at the graph, we write out the answer: for and the equation has two solutions; for , the equation has one solution; for , the equation has no solutions.
