Formula y = k/ x, the graph is a hyperbola. In Part 1 GIA this function offered without displacements along the axes. Therefore it has only one parameter k. The most big difference in appearance graphics depends on sign k.
It is more difficult to see differences in graphs if k one character:
As we 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 manual for preparing for the State Examination, is simply a “masterpiece”:
Not only that, in a fairly small picture, closely spaced graphs simply merge. Also, hyperbolas with positive and negative k are depicted in one coordinate plane. Which will completely disorient anyone who looks at this drawing. The “cool little star” just catches your eye.
Thank God this is just a training task. IN real options more correct formulations and obvious drawings were proposed.
Let's figure out how to determine the coefficient k according to the graph of the function.
From the formula: y = k/x it 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.
Therefore the formula of 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,
On this chart it is impossible to find a single integer point. Therefore the value k can be determined very approximately.
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 hyperbola is located in the 1st and 3rd coordinate angles(quadrants),
k < 0 - во 2-м и 4-ом.
If k modulo greater than 1 ( k= 2 or k= - 2), then the graph is located above 1 (below - 1) along the y-axis and looks wider.
If 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” towards zero:
1. Fractional linear function and her schedule
A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.
With the concept rational numbers you probably already know each other. Likewise rational functions are functions that can be represented as the quotient of two polynomials.
If a fractional rational function is the quotient of two linear functions - polynomials of the first degree, i.e. function of the form
y = (ax + b) / (cx + d), then it is called fractional linear.
Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is constant ). Fractional linear function defined for all real numbers except x = -d/c. Graphs of fractional linear functions do not differ in shape from the graph y = 1/x you know. A curve that is a graph of the function y = 1/x is called hyperbole. With an unlimited increase in x absolute value the function y = 1/x decreases indefinitely in absolute value and both branches of the graph approach the x-axis: the right one approaches from above, and the left one from below. The lines to which the branches of a hyperbola approach are called its asymptotes.
Example 1.
y = (2x + 1) / (x – 3).
Solution.
Let's select the whole part: (2x + 1) / (x – 3) = 2 + 7/(x – 3).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segment to the right, stretching along the Oy axis 7 times and shifting 2 unit segments upward.
Any fraction y = (ax + b) / (cx + d) can be written in a similar way, highlighting the “whole part”. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways along the coordinate axes and stretched along the Oy axis.
To construct a graph of any arbitrary fractional-linear function, it is not at all necessary to transform the fraction defining this function. Since we know that the graph is a hyperbola, it will be enough to find the straight lines to which its branches approach - the asymptotes of the hyperbola x = -d/c and y = a/c.
Example 2.
Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).
Solution.
The function is not defined, at x = -1. This means that the straight line x = -1 serves vertical asymptote. To find the horizontal asymptote, let’s find out what the values of the function y(x) approach when the argument x increases in absolute value.
To do this, divide the numerator and denominator of the fraction by x:
y = (3 + 5/x) / (2 + 2/x).
As x → ∞ the fraction will tend to 3/2. This means that the horizontal asymptote is the straight line y = 3/2.
Example 3.
Graph the function y = (2x + 1)/(x + 1).
Solution.
Let’s select the “whole part” of the fraction:
(2x + 1) / (x + 1) = (2x + 2 – 1) / (x + 1) = 2(x + 1) / (x + 1) – 1/(x + 1) =
2 – 1/(x + 1).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift by 1 unit to the left, a symmetrical display with respect to Ox and a shift by 2 unit segments up along the Oy axis.
Domain D(y) = (-∞; -1)ᴗ(-1; +∞).
Range of values E(y) = (-∞; 2)ᴗ(2; +∞).
Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases at each interval of the domain of definition.
Answer: Figure 1.
2. Fractional rational function
Let's consider fractionally rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than the first.
Examples of such rational functions:
y = (x 3 – 5x + 6) / (x 7 – 6) or y = (x – 2) 2 (x + 1) / (x 2 + 3).
If the function y = P(x) / Q(x) represents the quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complex, and it can sometimes be difficult to construct it accurately, with all the details. However, it is often enough to use techniques similar to those we have already introduced above.
Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы finite number elementary fractions, the form of which is determined by decomposing the denominator of the fraction Q(x) into the product of real factors:
P(x)/Q(x) = A 1 /(x – K 1) m1 + A 2 /(x – K 1) m1-1 + … + A m1 /(x – K 1) + …+
L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+
+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+
+ (M 1 x + N 1) / (x 2 +p t x + q t) m1 + … + (M m1 x + N m1) / (x 2 +p t x + q t).
Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.
Plotting graphs of fractional rational functions
Let's consider several ways to construct graphs of a fractional rational function.
Example 4.
Draw a graph of the function y = 1/x 2 .
Solution.
We use the graph of the function y = x 2 to construct a graph of y = 1/x 2 and use the technique of “dividing” the graphs.
Domain of definition D(y) = (-∞; 0)ᴗ(0; +∞).
Range of values E(y) = (0; +∞).
There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.
Answer: Figure 2.
Example 5.
Graph the function y = (x 2 – 4x + 3) / (9 – 3x).
Solution.
Domain D(y) = (-∞; 3)ᴗ(3; +∞).
y = (x 2 – 4x + 3) / (9 – 3x) = (x – 3)(x – 1) / (-3(x – 3)) = -(x – 1)/3 = -x/ 3 + 1/3.
Here we used the technique of factorization, reduction and reduction to a linear function.
Answer: Figure 3.
Example 6.
Graph the function y = (x 2 – 1)/(x 2 + 1).
Solution.
The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the ordinate. Before building a graph, let’s transform the expression again, highlighting the whole part:
y = (x 2 – 1)/(x 2 + 1) = 1 – 2/(x 2 + 1).
Note that isolating the integer part in the formula of a fractional rational function is one of the main ones when constructing graphs.
If x → ±∞, then y → 1, i.e. the line y = 1 is horizontal asymptote.
Answer: Figure 4.
Example 7.
Let's consider the function y = x/(x 2 + 1) and try to accurately find its largest value, i.e. the most high point right half graphics. To accurately construct this graph, today's knowledge is not enough. Obviously, our curve cannot “rise” very high, because the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, we need to solve the equation x 2 + 1 = x, x 2 – x + 1 = 0. This equation has no real roots. This means our assumption is incorrect. To find the most great value function, you need to find out at what largest A the equation A = x/(x 2 + 1) will have a solution. We will replace original equation square: Ax 2 – x + A = 0. This equation has a solution when 1 – 4A 2 ≥ 0. From here we find the largest value A = 1/2. 
Answer: Figure 5, max y(x) = ½.
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1. Fractional linear function and its graph
A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.
You are probably already familiar with the concept of rational numbers. Likewise rational functions are functions that can be represented as the quotient of two polynomials.
If a fractional rational function is the quotient of two linear functions - polynomials of the first degree, i.e. function of the form
y = (ax + b) / (cx + d), then it is called fractional linear.
Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is constant ). The linear fractional function is defined for all real numbers except x = -d/c. Graphs of fractional linear functions do not differ in shape from the graph y = 1/x you know. A curve that is a graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases unlimited in absolute value and both branches of the graph approach the abscissa: the right one approaches from above, and the left one from below. The lines to which the branches of a hyperbola approach are called its asymptotes.
Example 1.
y = (2x + 1) / (x – 3).
Solution.
Let's select the whole part: (2x + 1) / (x – 3) = 2 + 7/(x – 3).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretching along the Oy axis 7 times and shifting by 2 unit segments upward.
Any fraction y = (ax + b) / (cx + d) can be written in a similar way, highlighting the “whole part”. Consequently, the graphs of all fractional linear functions are hyperbolas, shifted in various ways along the coordinate axes and stretched along the Oy axis.
To construct a graph of any arbitrary fractional-linear function, it is not at all necessary to transform the fraction defining this function. Since we know that the graph is a hyperbola, it will be enough to find the straight lines to which its branches approach - the asymptotes of the hyperbola x = -d/c and y = a/c.
Example 2.
Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).
Solution.
The function is not defined, at x = -1. This means that the straight line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let’s find out what the values of the function y(x) approach when the argument x increases in absolute value.
To do this, divide the numerator and denominator of the fraction by x:
y = (3 + 5/x) / (2 + 2/x).
As x → ∞ the fraction will tend to 3/2. This means that the horizontal asymptote is the straight line y = 3/2.
Example 3.
Graph the function y = (2x + 1)/(x + 1).
Solution.
Let’s select the “whole part” of the fraction:
(2x + 1) / (x + 1) = (2x + 2 – 1) / (x + 1) = 2(x + 1) / (x + 1) – 1/(x + 1) =
2 – 1/(x + 1).
Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift by 1 unit to the left, a symmetrical display with respect to Ox and a shift by 2 unit segments up along the Oy axis.
Domain D(y) = (-∞; -1)ᴗ(-1; +∞).
Range of values E(y) = (-∞; 2)ᴗ(2; +∞).
Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases at each interval of the domain of definition.
Answer: Figure 1.
2. Fractional rational function
Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than first.
Examples of such rational functions:
y = (x 3 – 5x + 6) / (x 7 – 6) or y = (x – 2) 2 (x + 1) / (x 2 + 3).
If the function y = P(x) / Q(x) represents the quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complex, and it can sometimes be difficult to construct it accurately, with all the details. However, it is often enough to use techniques similar to those we have already introduced above.
Let the fraction be a proper fraction (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:
P(x)/Q(x) = A 1 /(x – K 1) m1 + A 2 /(x – K 1) m1-1 + … + A m1 /(x – K 1) + …+
L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+
+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+
+ (M 1 x + N 1) / (x 2 +p t x + q t) m1 + … + (M m1 x + N m1) / (x 2 +p t x + q t).
Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.
Plotting graphs of fractional rational functions
Let's consider several ways to construct graphs of a fractional rational function.
Example 4.
Draw a graph of the function y = 1/x 2 .
Solution.
We use the graph of the function y = x 2 to construct a graph of y = 1/x 2 and use the technique of “dividing” the graphs.
Domain of definition D(y) = (-∞; 0)ᴗ(0; +∞).
Range of values E(y) = (0; +∞).
There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.
Answer: Figure 2.
Example 5.
Graph the function y = (x 2 – 4x + 3) / (9 – 3x).
Solution.
Domain D(y) = (-∞; 3)ᴗ(3; +∞).
y = (x 2 – 4x + 3) / (9 – 3x) = (x – 3)(x – 1) / (-3(x – 3)) = -(x – 1)/3 = -x/ 3 + 1/3.
Here we used the technique of factorization, reduction and reduction to a linear function.
Answer: Figure 3.
Example 6.
Graph the function y = (x 2 – 1)/(x 2 + 1).
Solution.
The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the ordinate. Before building a graph, let’s transform the expression again, highlighting the whole part:
y = (x 2 – 1)/(x 2 + 1) = 1 – 2/(x 2 + 1).
Note that isolating the integer part in the formula of a fractional rational function is one of the main ones when constructing graphs.
If x → ±∞, then y → 1, i.e. the straight line y = 1 is a horizontal asymptote.
Answer: Figure 4.
Example 7.
Let's consider the function y = x/(x 2 + 1) and try to accurately find its largest value, i.e. the highest point on the right half of the graph. To accurately construct this graph, today's knowledge is not enough. Obviously, our curve cannot “rise” very high, because the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, we need to solve the equation x 2 + 1 = x, x 2 – x + 1 = 0. This equation has no real roots. This means our assumption is incorrect. To find the largest value of the function, you need to find out at what largest A the equation A = x/(x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 – x + A = 0. This equation has a solution when 1 – 4A 2 ≥ 0. From here we find the largest value A = 1/2.
Answer: Figure 5, max y(x) = ½.
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Home > LiteratureMunicipal educational institution
"Average secondary school No. 24"
Problematic – abstract work
on algebra and principles of analysis
Graphs of fractional rational functions
Pupils of the 11th grade A Natalia Sergeevna Tovchegrechko work supervisor Valentina Vasilievna Parsheva mathematics teacher, higher education teacher qualification category
Severodvinsk
Contents 3Introduction 4Main part. Graphs of fractional-rational functions 6 Conclusion 17 Literature 18Introduction
Plotting function graphs is one of the the most interesting topics V school math. 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. Such an ability to see both the formula and its geometric interpretation– 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 elementary functions. Only in the 11th grade did they learn to construct more complex functions using derivatives. When reading books:- N.A. Virchenko, I.I. Lyashko, K.I. Shvetsov. Directory. Function graphs. Kyiv “Naukova Dumka” 1979 V.S. Kramor. Repeat and systematize school course algebra and the beginnings of analysis. Moscow “Enlightenment” 1990 Yu.N. Makarychev, N.G. Mindyuk. Algebra - 8th grade. Additional chapters to school textbook. Moscow “Enlightenment”, 1998 I.M. Gelfand, E.G. Glagoleva, E.E. Shnol. Functions and graphs (basic techniques). Publishing house MCNMO, Moscow 2004 S.M. Nikolsky. M.K. Potapov, N.N. Reshetnikov, A.V. Shevkin. Algebra and beginnings of analysis: textbook for grade 11.
I saw that the graphs complex functions can be constructed without using a derivative, i.e. in elementary ways. Therefore, I chose the topic of my essay: “Graphs of 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. Function y=k/x on the set 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. When descending positive values argument (when x tends to zero), the values of the function 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 important role when 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.
, Where x is a variable, a,b, c, d – given numbers, and 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, с≠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 general view 
, 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 the 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 compose 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):
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 parallel in various ways 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 degree. 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 complex roots Q (x) multiplicity m 1 ... m t of a fraction of the form
Called elementary rational fractions the first, second, third and fourth types, respectively. Here A, B, C, k – 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 transfer along the x-axis by │k│ scale units to the right. Graph of a function of the form
It is easy to construct if you select in the denominator perfect square, 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 - natural number, can be performed by general scheme researching a function and plotting a graph in some specific examples you can successfully construct a graph by performing appropriate graph transformations; best way give methods 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 get the graph given function:
Figures 6, 7, 8 present examples of constructing function graphs 
And 
. Example 2. Graphing a Function 
:
(1); 
(2); 
(3); (4)
: 
(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 formWhere 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. This collection is the fifth issue prepared by the team of the Moscow City Pedagogical Gymnasium-Laboratory No. 1505 with the support of……. 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. Here the coefficients for X and the free terms in the numerator and denominator are given real numbers. The graph of a linear fractional function in the general case is hyperbola. The simplest fractional linear function y = - You- strikes inverse proportional relationship; the hyperbola representing it is well known from high school courses (Fig. 5.5). Rice. 5.5 Example. 5.3 Plot a graph of a linear fractional function: 2. In order to study the behavior of a function on the boundary of the domain of definition (i.e. when X-»3 and at X-> ±°°), it is useful to convert this expression to the sum of two terms as follows: Since the first term is constant, the behavior of the function on the boundary is actually determined by the second, variable term. Having studied the process of its change, when X->3 and X->±°°, we do the following conclusions relative to a given function: Comment. The information obtained in this section is the most important for characterizing the behavior of the graph of a function in the remote part of the plane (figuratively speaking, at infinity). perbola intersects the axis Oh at the point M x = (0;-^). draw a hyperbole graphically reflecting the properties of this function (Fig. 5.6). Rice. 5.6 The functions discussed up to this point are called algebraic. Let's now move on to consider transcendental functions.
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