The methodical material is for reference purposes and covers a wide range of topics. The article provides an overview of the graphs of the main elementary functions and considers the most important issue - how to correctly and FAST build a graph. In the course of studying higher mathematics without knowing the graphs of the basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, to remember some function values. We will also talk about some properties of the main functions.
I do not pretend to completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, on practice - those things with which one has to face literally at every step, in any topic of higher mathematics. Charts for dummies? You can say so.
By popular demand from readers clickable table of contents:
In addition, there is an ultra-short abstract on the topic
– master 16 types of charts by studying SIX pages!
Seriously, six, even I myself was surprised. This abstract contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!
And we start right away:
How to build coordinate axes correctly?
In practice, tests are almost always drawn up by students in separate notebooks, lined in a cage. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for the high-quality and accurate design of the drawings.
Any drawing of a function graph starts with coordinate axes.
Drawings are two-dimensional and three-dimensional.
Let us first consider the two-dimensional case Cartesian coordinate system:
1) We draw coordinate axes. The axis is called x-axis , and the axis y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo's beard.
2) We sign the axes with capital letters "x" and "y". Don't forget to sign the axes.
3) Set the scale along the axes: draw zero and two ones. When making a drawing, the most convenient and common scale is: 1 unit = 2 cells (drawing on the left) - stick to it if possible. However, from time to time it happens that the drawing does not fit on a notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). Rarely, but it happens that the scale of the drawing has to be reduced (or increased) even more
DO NOT scribble from a machine gun ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two units along the axes. Sometimes instead of units, it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely set the coordinate grid.
It is better to estimate the estimated dimensions of the drawing BEFORE the drawing is drawn.. So, for example, if the task requires drawing a triangle with vertices , , , then it is quite clear that the popular scale 1 unit = 2 cells will not work. Why? Let's look at the point - here you have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale 1 unit = 1 cell.
By the way, about centimeters and notebook cells. Is it true that there are 15 centimeters in 30 notebook cells? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. It may seem like nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automotive industry, falling planes or exploding power plants.
Speaking of quality, or a brief recommendation on stationery. To date, most of the notebooks on sale, without saying bad words, are complete goblin. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! Save on paper. For clearance control works I recommend using the notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, cage) or Pyaterochka, although it is more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smears or tears paper. The only "competitive" ballpoint pen in my memory is "Erich Krause". She writes clearly, beautifully and stably - either with a full stem, or with an almost empty one.
Additionally: the vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Vector basis, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.
3D case
It's almost the same here.
1) We draw coordinate axes. Standard: applicate axis – directed upwards, axis – directed to the right, axis – downwards to the left strictly at an angle of 45 degrees.
2) We sign the axes.
3) Set the scale along the axes. Scale along the axis - two times smaller than the scale along the other axes. Also note that in the right drawing, I used a non-standard "serif" along the axis (this possibility has already been mentioned above). From my point of view, it’s more accurate, faster and more aesthetically pleasing - you don’t need to look for the middle of the cell under a microscope and “sculpt” the unit right up to the origin.
When doing a 3D drawing again - give priority to scale
1 unit = 2 cells (drawing on the left).
What are all these rules for? Rules are there to be broken. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect from the point of view correct design. I could draw all the graphs by hand, but it’s really scary to draw them, as Excel is reluctant to draw them much more accurately.
Graphs and basic properties of elementary functions
The linear function is given by the equation . Linear function graph is direct. In order to construct a straight line, it is enough to know two points.
Example 1
Plot the function. Let's find two points. It is advantageous to choose zero as one of the points.
If , then
We take some other point, for example, 1.
If , then
When preparing tasks, the coordinates of points are usually summarized in a table:
And the values themselves are calculated orally or on a draft, calculator.
Two points are found, let's draw:
When drawing up a drawing, we always sign the graphics.
It will not be superfluous to recall special cases of a linear function:
Notice how I placed the captions, signatures should not be ambiguous when studying the drawing. AT this case it was extremely undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.
1) A linear function of the form () is called direct proportionality. For example, . The direct proportionality graph always passes through the origin. Thus, the construction of a straight line is simplified - it is enough to find only one point.
2) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is built immediately, without finding any points. That is, the entry should be understood as follows: "y is always equal to -4, for any value of x."
3) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also built immediately. The entry should be understood as follows: "x is always, for any value of y, equal to 1."
Some will ask, well, why remember the 6th grade?! That's how it is, maybe so, only during the years of practice I met a good dozen students who were baffled by the task of constructing a graph like or .
Drawing a straight line is the most common action when making drawings.
The straight line is discussed in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a straight line on a plane.
Quadratic function graph, cubic function graph, polynomial graph
Parabola. Graph of a quadratic function 
() is a parabola. Consider the famous case:
Let's recall some properties of the function.
So, the solution to our equation: - it is at this point that the vertex of the parabola is located. Why this is so can be learned from the theoretical article on the derivative and the lesson on the extrema of the function. In the meantime, we calculate the corresponding value of "y":
So the vertex is at the point
Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function 
– is not even, but, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can be figuratively called a "shuttle" or the "back and forth" principle with Anfisa Chekhova.
Let's make a drawing:
From the considered graphs, another useful feature comes to mind:
For a quadratic function 
() the following is true:
If , then the branches of the parabola are directed upwards.
If , then the branches of the parabola are directed downwards.
In-depth knowledge of the curve can be obtained in the lesson Hyperbola and parabola.
The cubic parabola is given by the function . Here is a drawing familiar from school:
We list the main properties of the function
Function Graph
It represents one of the branches of the parabola. Let's make a drawing:
The main properties of the function:
In this case, the axis is vertical asymptote for the hyperbola graph at .
It will be a BIG mistake if, when drawing up a drawing, by negligence, you allow the graph to intersect with the asymptote.
Also one-sided limits, tell us that a hyperbole not limited from above and not limited from below.
Let's explore the function at infinity: , that is, if we start to move along the axis to the left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.
So the axis is horizontal asymptote for the graph of the function, if "x" tends to plus or minus infinity.
The function is odd, which means that the hyperbola is symmetrical with respect to the origin. This fact is obvious from the drawing, in addition, it can be easily verified analytically: 
.
The graph of a function of the form () represents two branches of a hyperbola.
If , then the hyperbola is located in the first and third coordinate quadrants(see picture above).
If , then the hyperbola is located in the second and fourth coordinate quadrants.
It is not difficult to analyze the specified regularity of the place of residence of the hyperbola from the point of view of geometric transformations of graphs.
Example 3
Construct the right branch of the hyperbola
We use the pointwise construction method, while it is advantageous to select the values so that they divide completely:
Let's make a drawing:
It will not be difficult to construct the left branch of the hyperbola, here the oddness of the function will just help. Roughly speaking, in the pointwise construction table, mentally add a minus to each number, put the corresponding dots and draw the second branch.
Detailed geometric information about the considered line can be found in the article Hyperbola and parabola.
Graph of an exponential function
In this paragraph, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponent that occurs.
I remind you that - this is an irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points is probably enough:
Let's leave the graph of the function alone for now, about it later.
The main properties of the function:
Fundamentally, the graphs of functions look the same, etc.
I must say that the second case is less common in practice, but it does occur, so I felt it necessary to include it in this article.
Graph of a logarithmic function
Consider a function with natural logarithm .
Let's do a line drawing:
If you forgot what a logarithm is, please refer to school textbooks.
The main properties of the function:
Domain: 
Range of values: .
The function is not limited from above: 
, albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: 
. So the axis is vertical asymptote
for the graph of the function with "x" tending to zero on the right.
Be sure to know and remember the typical value of the logarithm: .
Fundamentally, the plot of the logarithm at the base looks the same: , , (decimal logarithm to base 10), etc. At the same time, the larger the base, the flatter the chart will be.
We will not consider the case, something I don’t remember when the last time I built a graph with such a basis. Yes, and the logarithm seems to be a very rare guest in problems of higher mathematics.
In conclusion of the paragraph, I will say one more fact: Exponential Function and Logarithmic Functionare two mutual inverse functions . If you look closely at the graph of the logarithm, you can see that this is the same exponent, just it is located a little differently.
Graphs of trigonometric functions
How does trigonometric torment begin at school? Correctly. From the sine
Let's plot the function
This line is called sinusoid.
I remind you that “pi” is an irrational number:, and in trigonometry it dazzles in the eyes.
The main properties of the function:
This function is periodical with a period. What does it mean? Let's look at the cut. To the left and to the right of it, exactly the same piece of the graph repeats endlessly.
Domain: , that is, for any value of "x" there is a sine value.
Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.
The function is written in general form as y = or f(x) =
y and x are inversely proportional quantities, i.e. when one increases, the other decreases (check by plugging the numbers into the function)
Unlike the previous function, in which x 2 always produces positive values, here we cannot say that - = , since these would be completely opposite numbers. Such functions are called odd.
For example, let's build a graph y =
Naturally, x cannot be equal to zero (x ≠ 0)
branches hyperbolas lie in the 1st and 3rd parts of the coordinates.
They can infinitely approach the abscissa and ordinate axes and never reach them, even if "x" becomes equal to a billion. The hyperbola will be infinitely close, but still will not intersect with the axes (such is the mathematical sadness).
Let's build a graph for y = -
And now the branches of the hyperbola are in the second and 4th quarters of the coordinate plane.
As a result, complete symmetry can be observed between all branches.
If a constant is added to the ARGUMENT of the function, then a shift (parallel translation) of the graph along the axis occurs. Consider a function and a positive number :
Rules:
1) to build a function graph, you need to shift the graph ALONG axes per units to the left;
2) to build a function graph, you need to shift the graph ALONG axes per units right.
Example 6
Plot a function
We take a parabola and shift it along the x-axis by 1 unit right:
The “identification beacon” is the value, this is where the top of the parabola is located.
Now, I think no one will have any difficulties with plotting the graph (demonstration example of the beginning of the lesson) - the cubic parabola needs to be shifted 2 units to the left.
Here is another typical case:
Example 7
Plot a function
Let's shift the hyperbola (black color) along the axis by 2 units to the left:
Moving the hyperbola "gives out" a value that is not included in function scope. AT this example, and straight line equation sets vertical asymptote(red dotted line) graph of the function (red solid line). Thus, with parallel translation, the asymptote of the graph also shifts (which is obvious).
Let's get back to trigonometric functions:
Example 8
Plot a function
The sine graph (black color) will be shifted along the axis along the axis by to the left:
Let's take a closer look at the resulting red graph .... That's exactly the cosine plot! In fact, we got a geometric illustration reduction formulas, and before you, perhaps, the most "famous" formula connecting these trigonometric functions. The graph of the function is obtained by shifting the sinusoid along the axis by units to the left (as already mentioned in the lesson Graphs and properties of elementary functions). Similarly, one can verify the validity of any other reduction formulas.
Consider the composition rule when the argument is a linear function: , while the parameter "ka" not equal zero or one, the parameter "be" - not equal zero. How to plot such a function? From the school course, we know that multiplication has priority over addition, therefore, it would seem that at first we compress / stretch / display the graph depending on the value, and then we shift it by units. But there is a pitfall here, and the correct algorithm is as follows:
The function argument must be presented in the form and sequentially perform the following transformations:
1) The graph of the function is compressed (or stretched) to the axis (from the axis) of the ordinates: (if , then the graph should additionally be displayed symmetrically with respect to the axis).
2) The graph of the resulting function is shifted to the left (or right) along the x-axis on the (!!!) units, as a result of which the desired graph will be built.
Example 9
Plot a function
Let's represent the function in the form and perform the following transformations: a sinusoid (black color):
1) squeeze to the axis twice: (blue color);
2) move along the axis on the (!!!) to the left: (Red color):
The example seems to be simple, but flying with a parallel transfer is easier than a breeze. The graph shifts by , not by .
We continue to deal with the functions of the beginning of the lesson:
Example 10
Plot a function
Let's represent the function as . In this case: The construction will be carried out in three steps. Graph of the natural logarithm:
1) squeeze to the axis 2 times: ;
2) display symmetrically relative to the axis : ;
3) move along the axis on the (!!!) to the right: :
For self-control, you can substitute a pair of X values into the final function, for example, and check the resulting graph.
In the considered paragraphs, the events took place "horizontally" - the accordion plays, the legs dance to the left / right. But similar transformations occur in the "vertical" direction - along the axis. The fundamental difference is that they are connected not with the ARGUMENT, but with the FUNCTION ITSELF.
Stretching (compressing) the graph ALONG the y-axis.
Symmetric display of the graph relative to the abscissa axis
The structure of the second part of the article will be very similar.
1) If the FUNCTION is multiplied by a number, then stretching its graph along the y-axis.
rule stretch along the axis in time.
2) If the FUNCTION is multiplied by a number, then compression of its graph along the y-axis.
rule: to plot the function , where , you need the graph of the function shrink along the axis in time.
Guess which function I'll be trying again =)
Example 11
Build function graphs.
We take a sinusoid by the crown / heels:
And draw out her along the axis 2 times:
The period of the function has not changed and is , but the values (all but zero) have increased modulo twice, which is logical - after all, the function is multiplied by 2, and the range of its values is doubled: .
Now compress sinusoid along the axis 2 times:
Similarly, the period has not changed, but the range of the function has "flattened" twice: .
No, I don’t have any bias towards the sinusoid, I just wanted to demonstrate how the function graphs (Examples No. 1,3) differ from the newly built counterparts. Try again to analyze and better understand these elementary cases. Even minimal knowledge of graph transformations will provide you with invaluable help in solving other problems of higher mathematics! . cases
Function Coefficient k can take any value except k = 0. Let us first consider the case when k = 1; Thus, first we will talk about the function.
To build a graph of the function, we will do the same as in the previous paragraph: we will give the independent variable x several specific values and calculate (using the formula) the corresponding values of the dependent variable y. True, this time it is more convenient to carry out calculations and constructions gradually, first giving the argument only positive values, and then only negative ones.
First stage. If x \u003d 1, then y \u003d 1 (recall that we use the formula);
Second phase.
In short, we have compiled the following table:
And now let's combine the two stages into one, i.e., from two figures 24 and 26 we will make one (Fig. 27). That's what it is function graph it is called hyperbole.
Let's try to describe the geometric properties of the hyperbola according to the drawing.
Firstly, we notice that this line looks as beautiful as a parabola, because it has symmetry. Any line passing through the origin O and located at the first and third coordinate angles intersects the hyperbola at two points that lie on this line on opposite sides of the point O, but at equal distances from it (Fig. 28). This is inherent, in particular, to the points (1; 1) and (- 1; - 1),
And so on. So - About the center of symmetry of the hyperbola. It is also said that the hyperbola is symmetrical with respect to the origin coordinates.
Secondly, we see that the hyperbola consists of two parts symmetrical about the origin; these are commonly referred to as branches of the hyperbola.
Thirdly, we notice that each branch of the hyperbola in one direction comes closer and closer to the abscissa axis, and in the other direction - to the ordinate axis. In such cases, the corresponding lines are called asymptotes.
Hence, the graph of the function , i.e. hyperbola has two asymptotes: the x-axis and the y-axis.
If you carefully analyze the constructed graph, you can find another geometric property that is not as obvious as the previous three (mathematicians usually say this: “a more subtle property”). A hyperbola has not only a center of symmetry, but also axes of symmetry.
Indeed, let's construct a straight line y = x (Fig. 29). Now look: dots 
located on opposite sides of the straight, but at equal distances from it. They are symmetrical about this line. The same can be said about points, where, of course, this means that the line y \u003d x is the axis of symmetry of the hyperbola (as well as y \u003d -x)
Example 1. Find the smallest and largest values of the function a) on the segment ; b) on the segment [- 8, - 1].
Solution, a) Let's build a graph of the function and select that part of it that corresponds to the values of the variable x from the segment (Fig. 30). For the selected part of the graph, we find:
b) Construct a graph of the function and select that part of it that corresponds to the values of the variable x from segment[- 8, - 1] (Fig. 31). For the selected part of the graph, we find:
So, we have considered the function for the case when k= 1. Now let k be a positive number other than 1, for example k = 2.
Let's consider a function and make a table of values of this function:
Construct points (1; 2), (2; 1), (-1; -2), (-2; -1),
on the coordinate plane (Fig. 32). They outline some line, consisting of two branches; we will carry it out (Fig. 33). Like the graph of a function, this line is called a hyperbola.
Consider now the case when k< 0; пусть, например, k = - 1. Построим график функции (здесь k = - 1).
In the previous paragraph, we noted that the graph of the function y \u003d -f (x) is symmetrical to the graph of the function y \u003d f (x) about the x-axis. In particular, this means that the graph of the function y \u003d - f (x) is symmetrical to the graph of the function y \u003d f (x) about the x axis. In particular, this means that schedule, is symmetrical to the graph with respect to the abscissa axis (Fig. 34) Thus, we get a hyperbola, the branches of which are located in the second and fourth coordinate angles.
In general, the graph of the function 
is a hyperbola whose branches are located in the first and third coordinate angles if k > 0 (Fig. 33), and in the second and fourth coordinate angles if k< О (рис. 34). Точка (0; 0) - центр симметрии гиперболы, оси координат - асимптоты гиперболы.
It is common to say that two quantities x and y are inversely proportional if they are related by the relation xy = k (where k is a number other than 0), or equivalently, . For this reason, the function is sometimes called inverse proportionality (by analogy with the function y - kx, which, as you probably
remember, called direct proportionality); the number k is the coefficient of the inverse proportionality.
Function properties for k > 0
Describing the properties of this function, we will rely on its geometric hyperbola model (see Fig. 33).
2. y > 0 for x>0; y<0 при х<0.
3. The function decreases on the intervals (-°°, 0) and (0, +°°).
5. Neither the smallest nor the largest values of the function
Function properties for k< 0
Describing the properties of this function, we will rely on its geometric model- hyperbole (see Fig. 34).
1. The domain of the function consists of all numbers except x = 0.
2. y > 0 at x< 0; у < 0 при х > 0.
3. The function increases on the intervals (-oo, 0) and (0, +oo).
4. The function is not limited from below or from above.
5. The function has neither the smallest nor the largest values.
6. The function is continuous on the intervals (-oo, 0) and (0, +oo) and undergoes a discontinuity at x = 0.
Lesson content lesson summary support frame lesson presentation accelerative methods interactive technologies Practice tasks and exercises self-examination workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures graphics, tables, schemes humor, anecdotes, jokes, comics parables, sayings, crossword puzzles, quotes Add-ons abstracts articles chips for inquisitive cheat sheets textbooks basic and additional glossary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in the textbook elements of innovation in the lesson replacing obsolete knowledge with new ones Only for teachers perfect lessons calendar plan for a year guidelines discussion programs Integrated LessonsHello, dear students of Argemony University! I welcome you to another lecture on the magic of functions and integrals.
Today we will talk about hyperbole. Let's start simple. The simplest form of a hyperbola is:
This function, in contrast to the straight line in its standard forms, has a singularity. As we know, the denominator of a fraction cannot be equal to zero, because you cannot divide by zero.
x ≠ 0
From this we conclude that the domain of definition is the entire real line, except for the point 0: (-∞; 0) ∪ (0; +∞).
If x tends to 0 from the right (written like this: x->0+), i.e. becomes very, very small, but still positive, then y becomes very, very large positive (y->+∞).
If x tends to 0 from the left (x->0-), i.e. becomes very, very small in absolute value, but remains negative, then y will also be negative, but in absolute value it will be very large (y->-∞).
If x tends to plus infinity (x->+∞), i.e. becomes a very large positive number, then y will become more and more smaller positive number, i.e. will tend to 0, remaining positive all the time (y->0+).
If x tends to minus infinity (x->-∞), i.e. becomes a large modulo, but a negative number, then y will also always be a negative number, but a small modulo (y->0-).
Y, like x, cannot take on the value 0. It only tends to zero. Therefore, the set of values is the same as the domain of definition: (-∞; 0) ∪ (0; +∞).
Based on these considerations, we can schematically draw a graph of the function
It can be seen that the hyperbola consists of two parts: one is in the 1st coordinate corner, where the x and y values are positive, and the second part is in the third coordinate corner, where the x and y values are negative.
If we move from -∞ to +∞, then we see that our function decreases from 0 to -∞, then there is a sharp jump (from -∞ to +∞) and the second branch of the function begins, which also decreases, but from +∞ to 0. That is, this hyperbola is decreasing.
If you change the function just a little: use the minus magic,
(1")
Then the function miraculously moves from the 1st and 3rd quarters to the 2nd and 4th quarters and becomes increasing.
Recall that the function is increasing, if for two values x 1 and x 2 such that x 1<х 2 , значения функции находятся в том же отношении f(х 1) < f(х 2).
And the function will be waning if f(x 1) > f(x 2) for the same values of x.
The branches of the hyperbola approach the axes, but never cross them. Such lines, which the graph of a function approaches, but never crosses, are called asymptote this function.
For our function (1), the asymptotes are the straight lines x=0 (OY axis, vertical asymptote) and y=0 (OX axis, horizontal asymptote).
Now let's complicate the simplest hyperbole a little and see what happens to the graph of the function.
(2)
Just added the constant "a" to the denominator. Adding some number to the denominator as a term to x means moving the entire "hyperbolic construction" (together with the vertical asymptote) by (-a) positions to the right if a is a negative number, and by (-a) positions to the left if a is a positive number.
On the left graph, a negative constant is added to x (a<0, значит, -a>0), which causes the chart to move to the right, and on the right chart, a positive constant (a>0), due to which the chart is moved to the left.
And what kind of magic can affect the transfer of the "hyperbolic construction" up or down? Adding a constant term to a fraction.
(3)
Now our entire function (both branches and the horizontal asymptote) will go up b positions if b is a positive number, and go down b positions if b is a negative number.
Please note that the asymptotes move along with the hyperbola, i.e. the hyperbola (both of its branches) and both of its asymptotes must necessarily be considered as an inseparable construction that moves as one to the left, right, up or down. It's a very pleasant feeling when you can make the whole function move in any direction by just adding some number. Why not magic, which you can master very easily and direct it at your discretion in the right direction?
By the way, you can control the movement of any function in this way. In the next lessons, we will consolidate this skill.
Before asking you homework, I want to draw your attention to this function
(4)
The lower branch of the hyperbola moves upward from the 3rd coordinate angle to the second one, to the angle where the value of y is positive, i.e. this branch is reflected symmetrically about the OX axis. And now we get an even function.
What does "even function" mean? The function is called even, if the condition is met: f(-x)=f(x)
The function is called odd, if the condition is met: f(-x)=-f(x)
In our case
(5)
Every even function is symmetric with respect to the OY axis, i.e. the parchment with the drawing of the graph can be folded along the OY axis, and the two parts of the graph will exactly match each other.
As you can see, this function also has two asymptotes - horizontal and vertical. Unlike the functions considered above, this function is increasing on one of its parts, and decreasing on the other.
Let's try to guide this graph now by adding constants.
(6)
Recall that adding a constant as a term to "x" causes the entire graph (together with the vertical asymptote) to move horizontally, along the horizontal asymptote (to the left or right, depending on the sign of this constant).
(7)
And adding the constant b as a term to the fraction causes the graph to move up or down. Everything is very simple!
Now try experimenting with this magic yourself.
Homework 1.
Everyone takes two functions for their experiments: (3) and (7).
a = the first digit of your LD
b=second digit of your LD
Try to get to the magic of these functions, starting with the simplest hyperbole, as I did in the lesson, and gradually adding your own constants. Function (7) can already be modeled based on the final form of function (3). Specify domains of definition, set of values, asymptotes. How functions behave: decrease, increase. Even odd. In general, try to conduct the same research as it was in the lesson. You may find something else that I forgot to mention.
By the way, both branches of the simplest hyperbola (1) are symmetrical with respect to the bisector 2 and 4 of the coordinate angles. Now imagine that the hyperbola began to rotate around this axis. We get just such a nice figure, which can be used.
Task 2. Where can this figure be used? Try to draw a figure of rotation for the function (4) about its axis of symmetry and discuss where such a figure can be used.
Remember how we got a straight line with a point punched out at the end of the last lesson? And here is the last task 3.
Create a graph for this function:
(8)
The coefficients a, b are the same as in task 1.
c=3rd digit of your LD or a-b if your LD is two digits.
A little hint: first, the fraction obtained after substituting the numbers must be simplified, and then you will get the usual hyperbola, which you need to build, but at the end you need to take into account the domain of the original expression.