A fractionally linear function has its properties. Problem-based abstract work

Let's consider the questions of methodology for studying such a topic as “constructing a graph of a fractional linear function.” Unfortunately, its study has been removed from the basic program and the math tutor in his classes does not touch on it as often as we would like. However, no one has canceled math classes yet, nor have they canceled the second part of the GIA. And in the Unified State Examination there is a possibility of its penetration into the body of task C5 (through 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 mathematics tutor develops methods of explanation for the main sections of the school curriculum during the first 5-7 years of work. 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, quitters and truants to purposeful talents.

Over time, a math tutor develops the skill of explaining complex concepts in simple language without sacrificing mathematical completeness and accuracy. An individual style of presentation of material, speech, visual accompaniment and recording 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 the right words and notes, examples for the beginning of the lesson, for the middle and the end, as well as correctly compose exercises for homework.

Some particular techniques for working with the theme will be discussed in this article.

What graphs does a math tutor start with?

You need to start by defining the concept being studied. Let me remind you that a fractional linear function is a function of the form . Its construction comes down to building the most common hyperbole using well-known simple techniques for transforming graphs. In practice, they turn out to be simple only for the tutor himself. Even if a strong student comes to the teacher, with sufficient speed of calculations and transformations, he still has to teach these techniques separately. Why? At school in the 9th grade, graphs are constructed only by shifting and do not use methods of adding numerical multipliers (compression and stretching methods). What graph does a math tutor use? Where is the best place to start? All preparation is carried out using the example of the most convenient, in my opinion, function . What else should I use? Trigonometry in the 9th grade is studied without graphs (and in textbooks that have been modified to suit the conditions of the State Examination in Mathematics, they are not taught at all). The quadratic function does not have the same “methodological weight” in this topic as the root does. Why? In grade 9, the quadratic trinomial is studied in detail and the student is quite capable of solving construction problems without shifts. The form instantly evokes a reflex to open the brackets, after which you can apply the rule of standard plotting through the vertex of a parabola and a table of values. With such a maneuver it will not be possible to perform and it will be easier for a mathematics tutor to motivate the student to study general transformation techniques. Using the module 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 included in the number of transformations being studied.

So, the tutor has nothing left more convenient and effective than to prepare for transformations using the square root. You need practice in constructing graphs of something like this. Let us consider that this preparation was a great success. The child can move and even compress/stretch graphs. What's next?

The next stage is learning to isolate a whole part. Perhaps this is the main task of a mathematics tutor, because after the whole part is allocated, it takes on the lion's share of the entire computational load on the topic. It is extremely important to prepare the function in 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 manner, and on the other hand, mathematically precise and harmonious.

Let me remind you that to build a graph you need to convert the fraction to the form . Precisely for this, and not for
, keeping the denominator. Why? It is difficult to perform transformations on a graph that 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, you can’t immediately figure out which points to connect. Therefore, compressing or stretching a hyperbole is extremely inconvenient. A math tutor is simply obliged to teach a student how to make do with shifts alone.

To do this, in addition to selecting the whole part, you also need to remove the coefficient from the denominator c.

Selecting the integer part from a fraction

How to teach highlighting a whole part? Mathematics tutors do not always adequately assess the student’s level of knowledge and, despite the absence in the program of a detailed study of the theorem on the division of polynomials with a remainder, they apply the rule of division by a corner. If a teacher takes on corner division, he will have to spend almost half of the lesson explaining it (if, of course, everything is carefully justified). Unfortunately, the tutor does not always have this time available. It’s better not to remember any corners at all.

There are two forms of working with a student:
1) The tutor shows him a ready-made algorithm using some example of a fractional function.
2) The teacher creates conditions for a logical search for this algorithm.

The implementation of the second path seems to me the most interesting for tutoring practice and extremely useful to develop student thinking. With the help of certain hints and directions, it is often possible to lead to the discovery of a certain sequence of correct steps. In contrast to the mechanical execution of a plan drawn up by someone, a 9th grade student learns to look for it independently. Naturally, all explanations must be done with examples. For this purpose, let’s take a function 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 using a shift along the axes? Of course, the simultaneous presence of X in both the numerator and denominator. This means that it must be removed from the numerator. How to do this using identity transformations? There is only one way - to reduce the fraction. But we don't have equal factors (brackets). This means we need to try to create them artificially. But how? You can’t replace the numerator with the denominator without any identical transition. Let's try to transform the numerator so that it includes a parenthesis equal to the denominator. Let's put it there forcibly and “overlay” with coefficients so that when they “act” on the bracket, that is, when opening it and adding similar terms, a linear polynomial 2x+3 would be obtained.

The math tutor inserts gaps for coefficients in the form of empty rectangles (as textbooks for grades 5–6 often use) and sets the task to fill them in with numbers. The selection should be carried out from left to right, starting from the first pass. The student must imagine how he will open the bracket. Since its expansion will result in only one term with X, then its coefficient must 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 this can we move on to analyzing examples with an unpleasant appearance of the numerator and denominator (including fractional coefficients).

Go ahead. The teacher opens the bracket and signs the result directly above it.
You can shade the corresponding pair of factors. To the “opened term”, it is necessary to add such a number from the second gap in order to obtain the free coefficient of the old numerator. Obviously it's a 7.

Next, the fraction is broken down into the sum of individual fractions (I usually circle the fractions with a cloud, comparing their arrangement to the wings of a butterfly). And I say: “Let’s break the fraction with a butterfly.” Schoolchildren remember this phrase well.

The math tutor shows the whole process of isolating an entire part to a form to which you can already apply the hyperbola shift algorithm:

If the denominator has a leading coefficient that is not equal to one, then in no case should you leave it there. This will bring both the tutor and the student an extra headache associated with the need to carry out 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 work 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: All that remains is to move it to the right and 2 up.

If there is a “minus” between the whole part 2 and the remaining fraction, it is also better to include 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.

The golden rule of a math tutor:
all inconvenient coefficients that lead to symmetries, compression or stretching of the graph must be transferred to the numerator.

It is difficult to describe techniques for working with any topic. There is always a feeling of some understatement. To what extent we were able to talk about a fractional linear function is up to you to judge. Send your comments and reviews to the article (they can be written 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.

Home > Literature

Municipal educational institution

"Secondary school No. 24"

Problem-based abstract work

on algebra and principles of analysis

Graphs of fractional rational functions

Pupils of grade 11 A Tovchegrechko Natalya Sergeevna work supervisor Valentina Vasilievna Parsheva mathematics teacher, teacher of the highest qualification category

Severodvinsk

Contents 3Introduction 4Main part. Graphs of fractional-rational functions 6 Conclusion 17 Literature 18

Introduction

Graphing functions is one of the most interesting topics in school mathematics. One of the greatest mathematicians of our time, Israel Moiseevich Gelfand, wrote: “The process of constructing graphs is a way of transforming formulas and descriptions into geometric images. This graphing is a means of seeing formulas and functions and seeing how those functions change. For example, if it is written y=x 2, then you immediately see a parabola; if y=x 2 -4, you see a parabola lowered by four units; if y=4-x 2, then you see the previous parabola turned down. This ability to see both a formula and its geometric interpretation at once is important not only for studying mathematics, but also for other subjects. It is a skill that stays with you for life, like the ability to ride a bicycle, type or drive a car.” In mathematics lessons we build mainly the simplest graphs - graphs of elementary functions. Only in the 11th grade did they learn to construct more complex functions using derivatives. When reading books:

Purpose of the work: to study the relevant theoretical materials, to identify an algorithm for constructing graphs of fractional-linear and fractional-rational functions. Objectives: 1. formulate the concepts of fractional-linear and fractional-rational functions based on theoretical material on this topic; 2. find methods for constructing graphs of fractional-linear and fractional-rational functions.

Main part. Graphs of fractional rational functions

1. Fractional - linear function and its graph

We have already become familiar with a function of the form y=k/x, where k≠0, its properties and graph. Let's pay attention to one feature of this function. The function y=k/x on a set of positive numbers has the property that with an unlimited increase in the values of the argument (when x tends to plus infinity), the values of the functions, while remaining positive, tend to zero. As positive values of the argument decrease (when x tends to zero), the function values increase without limit (y tends to plus infinity). A similar picture is observed for the set of negative numbers. On the graph (Fig. 1), this property is expressed in the fact that the points of the hyperbola, as they move away to infinity (to the right or left, up or down) from the origin of coordinates, indefinitely approach the straight line: the x axis, when │x│ tends to plus infinity, or to the y-axis when │x│ tends to zero. This line is called asymptotes of the curve.

Rice. 1

The hyperbola y=k/x has two asymptotes: the x-axis and the y-axis. The concept of asymptote plays an important role in constructing graphs of many functions. Using the transformations of function graphs known to us, we can move the hyperbola y=k/x in the coordinate plane to the right or left, up or down. As a result, we will obtain new function graphs. Example 1. Let y=6/x. Let's shift this hyperbola to the right by 1.5 units, and then shift the resulting graph up 3.5 units. With this transformation, the asymptotes of the hyperbola y=6/x will also shift: the x axis will go into the straight line y=3.5, the y axis into the straight line y=1.5 (Fig. 2). The function whose graph we have plotted can be specified by the formula

Let's represent the expression on the right side of this formula as a fraction:

This means that Figure 2 shows a graph of the function given by the formula

This fraction has a numerator and denominator that are linear binomials with respect to x. Such functions are called fractional linear functions.

In general, a function defined by a formula of the form
, Where
x is a variable, a,b, c, d– given numbers, with c≠0 and
bc- ad≠0 is called a fractional linear function. Note that the requirement in the definition that c≠0 and
bc-ad≠0, significant. When c=0 and d≠0 or bc-ad=0 we get a linear function. Indeed, if c=0 and d≠0, then

If bc-ad=0, c≠0, expressing b from this equality through a, c and d and substituting it into the formula, we get:

So, in the first case we got a linear function of the general form
, in the second case – a constant
. Let us now show how to plot a linear fractional function if it is given by a formula of the form
Example 2. Let's plot the function
, i.e. let's present it in the form
: we select the whole part of the fraction, dividing the numerator by the denominator, we get:

So,
. We see that the graph of this function can be obtained from the graph of the function y=5/x using two successive shifts: shifting the hyperbola y=5/x to the right by 3 units, and then shifting the resulting hyperbola
up by 2 units. With these shifts, the asymptotes of the hyperbola y = 5/x will also move: the x axis 2 units up, and the y axis 3 units to the right. To construct a graph, we draw asymptotes in the coordinate plane with a dotted line: straight line y=2 and straight line x=3. Since the hyperbola consists of two branches, to construct each of them we will compile two tables: one for x<3, а другую для x>3 (i.e., the first one is to the left of the point of intersection of the asymptotes, and the second one is to the right of it):

By marking the points in the coordinate plane whose coordinates are indicated in the first table and connecting them with a smooth line, we obtain one branch of the hyperbola. Similarly (using the second table) we obtain the second branch of the hyperbola. The function graph is shown in Figure 3.

I like any fraction
can be written in a similar way, highlighting its entire part. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways parallel to the coordinate axes and stretched along the Oy axis.

Example 3.

Let's plot the function
.Since we know that the graph is a hyperbola, it is enough to find the straight lines to which its branches (asymptotes) approach, and a few more points. Let us first find the vertical asymptote. The function is not defined where 2x+2=0, i.e. at x=-1. Therefore, the vertical asymptote is the straight line x = -1. To find the horizontal asymptote, you need to look at what the function values approach when the argument increases (in absolute value), the second terms in the numerator and denominator of the fraction
relatively small. That's why

Therefore, the horizontal asymptote is the straight line y=3/2. Let's determine the intersection points of our hyperbola with the coordinate axes. At x=0 we have y=5/2. The function is equal to zero when 3x+5=0, i.e. at x = -5/3. Having marked the points (-5/3;0) and (0;5/2) on the drawing and drawing the found horizontal and vertical asymptotes, we will construct a graph (Fig. 4).

In general, to find the horizontal asymptote, you need to divide the numerator by the denominator, then y=3/2+1/(x+1), y=3/2 is the horizontal asymptote.

2. Fractional rational function

Consider the fractional rational function

In which the numerator and denominator are polynomials of the nth and mth degrees, respectively. Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и при том единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:Если:

Where k 1 ... k s are the roots of the polynomial Q (x), having, respectively, multiplicities m 1 ... m s, and the trinomials correspond to conjugation pairs of complex roots Q (x) of multiplicity m 1 ... m t fractions of the form

Called elementary rational fractions the first, second, third and fourth types, respectively. Here A, B, C, k are real numbers; m and m - natural numbers, m, m>1; a trinomial with real coefficients x 2 +px+q has imaginary roots. Obviously, the graph of a fractional-rational function can be obtained as the sum of graphs of elementary fractions. Graph of a function

We obtain from the graph of the function 1/x m (m~1, 2, ...) using parallel translation along the abscissa axis by │k│ scale units to the right. Graph of a function of the form

It is easy to construct if you select a complete square in the denominator, and then carry out the corresponding formation of the graph of the function 1/x 2. Graphing a Function

comes down to constructing the product of graphs of two functions:

y= Bx+ C And

Comment. Graphing a function

Where a d-b c 0 ,
,

where n is a natural number, it can be carried out according to the general scheme of studying a function and constructing a graph; in some specific examples, you can successfully construct a graph by performing the appropriate transformations of the graph; The best way is provided by the methods of higher mathematics. Example 1. Graph the function

Having isolated the whole part, we have

Fraction
Let's represent it as a sum of elementary fractions:

Let's build graphs of functions:

After adding these graphs, we obtain a graph of the given function:

Figures 6, 7, 8 present examples of constructing function graphs
And
. Example 2. Graphing a Function
:

(1);
(2);
(3); (4)

Example 3. Plotting the graph of a function

(1);
(2);
(3); (4)

Conclusion

When performing abstract work: - clarified her concepts of fractional-linear and fractional-rational functions: Definition 1. A linear fractional function is a function of the form , where x is a variable, a, b, c, and d are given numbers, with c≠0 and bc-ad≠0. Definition 2. A fractional rational function is a function of the form

Where n

Created an algorithm for plotting graphs of these functions;

Gained experience in plotting functions such as:

;

I learned to work with additional literature and materials, to select scientific information; - I gained experience in performing graphic work on a computer; - I learned how to write problem-based abstract work.

Annotation. On the eve of the 21st century, we were bombarded with an endless stream of talk and speculation about the information highway and the coming era of technology.

On the eve of the 21st century, we were bombarded with an endless stream of talk and speculation about the information highway and the coming era of technology.

Elective courses are one of the forms of organizing educational, cognitive and educational-research activities of high school students

Document

This collection is the fifth issue prepared by the team of the Moscow City Pedagogical Gymnasium-Laboratory No. 1505 with the support of…….

Mathematics and experience

Book

The paper attempts a large-scale comparison of different approaches to the relationship between mathematics and experience, which have developed mainly within the framework of apriorism and empiricism.

In this lesson we will look at the fractional linear function, solve problems using the fractional linear function, module, parameter.

Topic: Repetition

Lesson: Fractional linear function

Definition:

A function of the form:

For example:

Let us prove that the graph of this linear fractional function is a hyperbola.

Let's take the two out of brackets in the numerator and 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 propose a second method of proof, namely, divide the numerator by the denominator in a column:

Got:

It is important to be able to easily construct 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 graph of a function:

We have already converted this function and got:

To construct this graph, we will not shift the axes or the hyperbola itself. We use a standard method for constructing function graphs, using the presence of intervals of constant sign.

We act according to the algorithm. First, let's examine the given function.

Thus, we have three intervals of constant sign: on the far right () the function has a plus sign, then the signs alternate, since all roots have the first degree. So, on an interval the function is negative, on an interval the function is positive.

We construct a sketch of the graph in the vicinity of the roots and break points of the ODZ. We have: since at a point the sign of the function changes from plus to minus, 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 equal to zero, it means that when the value of the argument tends to three, the value of the fraction tends to infinity. In 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 leaves plus infinity.

Now we construct a sketch of the graph of the function in the vicinity of points at infinity, i.e. when the argument tends to plus or minus infinity. In this case, constant terms can be neglected. We have:

Thus, we have a horizontal asymptote and a vertical one, the center of the hyperbola is point (3;2). Let's illustrate:

Rice. 1. Graph of a hyperbola for example 1

Problems with a fractional linear function can be complicated by the presence of a modulus or parameter. To build, for example, a graph of the function, 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, parts of the graph located above the x-axis remain unchanged, and those located below the axis are mirrored relative to the x-axis. We get:

Rice. 3. Illustration for the algorithm

Example 2 - plot a function:

Rice. 4. Function graph for example 2

Consider the following task - construct a graph of the function. To do this, you must follow the following algorithm:

1. Graph the submodular function

Let's assume we get 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 argument values, no changes will occur. Regarding the second equation, we know that it is obtained by mapping it symmetrically about the y-axis. we have a graph of the function:

Rice. 6. Illustration for the algorithm

Example 3 - plot a function:

According to the algorithm, you first need to build a graph of the submodular function, we have already built it (see Figure 1)

Rice. 7. Graph of a function 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 going through all the values of the parameter and indicating 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 dissect the graph with a family of lines for different a, find the intersection points and write down the answer.

Looking at the graph, we write out the answer: when and the equation has two solutions; when the equation has one solution; when the equation has no solutions.

ax +b
A fractional linear function is a function of the form y = --- ,
cx +d

Where x– variable, a,b,c,d– some numbers, and c ≠ 0, ad -bc ≠ 0.

Properties of a fractional linear function:

The graph of a linear fractional function is a hyperbola, which can be obtained from the hyperbola y = k/x using parallel translations along the coordinate axes. To do this, the formula of the fractional linear function must be presented in the following form:

k
y = n + ---
x–m

Where n– the number of units by which the hyperbola shifts to the right or left, m– the number of units by which the hyperbola moves up or down. In this case, the asymptotes of the hyperbola are shifted to straight lines x = m, y = n.

An asymptote is a straight line to which the points of the curve approach as they move away to infinity (see the figure below).

As for parallel transfers, see the previous sections.

Example 1. Let's find the asymptotes of the hyperbola and plot the function:

x + 8
y = ---
x – 2

Solution:

k
Let's represent the fraction as n + ---
x–m

For this x+ 8 we write in the following form: x – 2 + 10 (i.e. 8 is represented as –2 + 10).

x+ 8 x – 2 + 10 1(x – 2) + 10 10
--- = ----- = ------ = 1 + ---
x – 2 x – 2 x – 2 x – 2

Why did the expression take this form? The answer is simple: do the addition (reducing both terms to a common denominator), and you will return to the previous expression. That is, this is the result of transforming a given expression.

So, we got all the necessary values:

k = 10, m = 2, n = 1.

Thus, we found the asymptotes of our hyperbola (based on the fact that x = m, y = n):

That is, one asymptote of the hyperbola runs parallel to the axis y at a distance of 2 units to the right of it, and the second asymptote runs parallel to the axis x at a distance of 1 unit above it.

Let's build a graph of this function. To do this we will do the following:

1) draw in the coordinate plane with a dotted line the asymptotes – the line x = 2 and the line y = 1.

2) since the hyperbola consists of two branches, then to construct these branches we will compile two tables: one for x<2, другую для x>2.

First, let's select the x values for the first option (x<2). Если x = –3, то:

10
y = 1 + --- = 1 – 2 = –1
–3 – 2

We choose arbitrarily other values x(for example -2, -1, 0 and 1). Calculate the corresponding values y. The results of all the calculations obtained are entered into the table:

Now let’s create a table for option x>2: