Another is sufficient sign of the foundation of the extremum. The growth and change of functions on intervals, extremums. Enough sign of extreme

The extremum point of a function is the point of the area of ​​function designation, in which the value of the function is set to the minimum or maximum value. The values ​​of the function at these points are called extrema (minimum and maximum) of the function.

Appointment. Krapka x1 areas of assigned function f(x) is called point of maximum function even though the value of the function at this point is greater than the value of the function at the points close to it, spreading right-handed and left-handed in it (to avoid unevenness f(x0 ) > f(x 0 + Δ x) x1 maximum.

Appointment. Krapka x2 areas of assigned function f(x) is called the minimum point of the function even though the value of the function at this point is less than the value of the function at the points close to it, the right-handed and angry in it (this is why the unevenness f(x0 ) < f(x 0 + Δ x) ). It seems to everyone that the function can be at the point x2 minimum.

Let's dot x1 - point of maximum function f(x). Todi in the interval up to x1 function grows This is similar to functions greater than zero ( f "(x) > 0 ), and in the interval after x1 the function changes, now, and similar functions less than zero ( f "(x) < 0 ). Тогда в точке x1

It is also possible that the point x2 - point to the minimum of the function f(x). Todi in the interval up to x2 the function changes, and the similar function is less than zero ( f "(x) < 0 ), а в интервале после x2 the function grows, and the similar function is greater than zero ( f "(x) > 0). Whose mind has the same point x2 Pokhіdna functions are equal to zero or not.

Fermat's theorem. What a point x0 - point of extremum of the function f(x) , then at the nth point the function is similar to zero ( f "(x) = 0) or not.

Appointment. Points, which have similar functions equal to zero or not, are called critical points .

example 1. Let's look at the function.

At the point x= 0 x= 0 is the critical point. However, as can be seen on the graph of the function, there is an increase in the entire area of ​​​​appointment, that's the point x= 0 is not an extremum of the function.

In this way, think about those that are worthy of a function to the point of reaching zero, or not necessary, or the necessary minds of an extremum, or not sufficient, you can point the shards and other applications of functions, for some of them, the mind can be conned, or else the function of an extremum. Tom mother needs sufficient signs, which allows you to judge, chi є in a specific critical point of extremum and yaky itself - maximum chi minimum.

Theorem (the first is sufficient sign of the basis of the extremum of the function). critical point x0 f(x) so that when passing through this point, the function changes the sign, moreover, if the sign changes from "plus" to "minus", then the maximum point, and if it changes from "minus" to "plus", then the minimum point.

How close is the point x0 , left-handed and right-handed in it, if it takes a sign, then it means that the function either changes, or only grows in the vicinity of the point x0 . In what direction at the point x0 there is no extremum.

Otzhe, to assign points to the extremum of the function, as needed :

1. Find a suitable function.
2. Set equal to zero and assign critical points.
3. Thoughts chi papers mark critical points on the numerical axis and mark the signs of a similar function subtracting intervals. If the sign changes from "plus" to "minus", then the critical point is the maximum point, and if it changes from "minus" to "plus", then the minimum point.
4. Calculate the value of the function at the extremum points.

butt 2. Know extremum functions .

Solution. We know the following functions:

It is equal to zero, in order to know the critical points:

.

So, if for any value of "ix" the banner is not equal to zero, then the number is equal to zero:

Take away one critical point x= 3. The sign of the opposite is significant in the intervals delimited by the point:

in the interval of minus inconsistency up to 3 - minus sign, so that the function changes,

in the interval of 3 to plus inconsistencies - a plus sign, so that the function grows.

Tobto, dot x= 3 - point minimum.

We know the value of the function at the minimum point:

In this order, the extremum point of the function is found: (3; 0), moreover, it is the minimum point.

Theorem (the other is enough sign of the basis of the extremum of the function). critical point x0 є extremum point of the function f(x); f ""(x) ≠ 0); f ""(x) > 0 ), then the point is the maximum, and the other way around is less than zero ( f ""(x) < 0 ), то точкой минимума.

Note 1. What is at the point x0 turn to zero and the first, and the other is dead, then in this point it is impossible to judge the manifestation of an extremum on the basis of another sufficient sign. It is necessary for this type of mood to be quickened by the first sufficient sign of the extremum of the function.

Respect 2. Another sufficient sign of the extremum of the function does not suffice, and even if in a stationary point the first is not good (there is no other bad). It is also necessary for this type of attitude to be quickened by the first sufficient sign of the extremum of the function.

Local nature of extremums of the function

It is obvious that the extremum of the function may have a local character - the value of the most and least of the values ​​of the function is equal to the closest values.

Let's say you look at your earnings at the time of the wedding one day. If you earned 45,000 rubles from the grass, and 42,000 rubles from the quarter, and 39,000 rubles from the red ones, then the grass earnings are the maximum of the earning function in terms of the nearest values. Ale have earned 71,000 rubles from yellow, 75,000 rubles from spring, and 74,000 rubles from leaf fall, so the same income - the minimum income function is equal to the closest values. You can easily bachite, so that the maximum average value of spring-grass-cherry is less than the minimum of spring-zhovtnya-leaf fall.

Speaking zagalneno, in the interim, the function may be the mother of a sprinkling of extremes, moreover, it may appear that the minimum of the function is greater than the maximum. So, for the function depicted a little more, .

So it is not necessary to think that the maximum and minimum of the function are, apparently, the largest and smallest values ​​on all parts that can be seen. At the point to the maximum, the function has the least value in the range of these values, if it is possible at all points, to reach the point close to the maximum, and at the point to the minimum - the least value in the range of these values, if it is close to the points to the minimum point.

Therefore, it can be clarified to better understand the point of the extremum of the function and call the points of the minimum the points of the local minimum, and the points of the maximum - the points of the local maximum.

Shukaemo extreme functions at once

example 3.

Solution. The function is assigned and without interruption on the whole number line. Її pokhіdna іsnuє also on the whole number line. Tom in to this particular type critical points є less ti, for yak, tobto. , stars that . Critical points and divide the entire area of ​​assigned function into three intervals of monotonicity: . Viberemo in the skin of them by one control point and we know the sign of the next one at the second point.

For an interval, a control point can be: known. Taking a point in the interval, we subtract, and taking a point in the interval, we can. Also, in intervals i , and in intervals . Zgіdno with the first sufficient sign of the extremum, at the point there is no extremum (shards are more likely to take the sign in the interval), and at the points the function can be minimum (the shards are less when passing through the next point, changing the sign from minus to plus). We know the relevant values ​​of the function: , a . At the interval, the function changes, the spikes at this interval, and the intervals increase, the spikes at that interval.

To clarify the future graphics, we know the points of the line of yoga with the coordinate axes. When we take equal , the root of which i , then two points (0; 0) and (4; 0) of the graph of the function are found. Vikoristovuyuchi all otrimani vіdomosti, budєmo schedule (div. on the cob butt).

For self-verification with rozrachunkah, you can speed up online similar calculator .

butt 4. Know the extremums of the function and induce the schedule.

The scope of the function is the whole number line, except for the points, tobto. .

For a quick follow-up, you can speed up the fact that the function of the steam room, shards . Therefore, the schedule is symmetrical about the axis Ouch that follow-up can only be used for the interval.

We know I'll go and critical points of the function:

1) ;

2) ,

But if the function knows the difference in this point, then it cannot be an extremum point.

in such a manner, function is set maє two critical points: i . Vrahovoyuchi pairing of functions, perevirim for another sufficient sign of extremum is only a point. For whom we know a friend I will die і significant її sign at: otrimaєmo. Since i , then є the minimum point of the function, at which .

In order to add more information about the schedule of the function, it is necessary to follow the behavior at the boundaries of the designated area:

(here the symbol indicates exercise x right-handed to zero, moreover x become overwhelmed with positive; similarly means exercise x to zero angry, moreover x become overwhelmed with negative). In such a rank, yakscho, then. Dali, we know

,

tobto. like that.

The break point with the axes of the graph function cannot be. Little one - on the cob butt.

For self-verification with rozrachunkah, you can speed up online similar calculator .

Prodovzhuєmo shukati extreme functions at once

Example 8. Know extremum functions.

Solution. We know the scope of the assigned function. So if nervousness can win over, then we are obsessed.

Let's know the first pokhіdnu functions.

duje important information about the behavior of the function, give rise to periods of growth and decay. Їхнє perebuvannya є part of the process follow-up functions and prompt graphics. Until then, extremum points, in which there is a change from growth to decline, or from a change to growth, are given special respect when value of the largest and smallest value of the function on the current interval.

In this article, there is a need to define, formulate a sufficient sign of an increase in that change in function over an interval and a sufficient reason for an extremum, we will put the whole theory to perfection by applying that task.

Navigation on the side.

The growth and change of the function on the interval.

Designated growing function.

The function y=f(x) grows on the interval X, as well as for whatever i nerіvnіst vykonuetsya. Otherwise, it seems - the greater value of the argument is greater than the value of the function.

Designated decay function.

The function y=f(x) changes by the interval X, as for any i nerіvnіst . Otherwise, apparently - the greater value of the argument is given by the lesser value of the function.

NOTE: as the function is assigned and without interruption in the intervals of growth or decay (a; b), then at x = a і x = b, then qi points are included in the interval of growth or decay. Don't overestimate the purpose of the growth and decay function for the interval X .

For example, from the powers of the basic elementary functions, we know that y=sinx is assigned and is uninterrupted by all the effective values ​​of the argument. Therefore, from the growth of the sine function on the intervals, we can confirm the growth of the sine function on the interval.

Krapki extremum, extremum functions.

Name the point maximum point functions y=f(x) , so all x in the neighborhood is fair. The value of the function at the point to the maximum is called function maximum i mean.

Name the point minimum point functions y=f(x) , so all x in the neighborhood is fair. The value of the function at the point of the minimum is called minimum function i mean.

Under the periphery of the point, understand the interval , de - Finish a small positive number.

The points of minimum and maximum are called extremum points, and the value of the function, which corresponds to the extremum points, is called function extrema.

Do not confuse extreme functions with the largest one lowest value functions.

On the first little one, the greatest value of the function on the top is reached at the point of maximum and the next maximum of the function, and on the other little one, the greatest value of the function is reached at the point x = b, but not at the point of maximum.

Enough to understand the growth of that changed function.

On the basis of sufficient minds (sign) of the growth of that changed function, there are gaps of the growth of that changed function.

The axis of the formula is a sign of the growth and change of the function on the interval:

• if a similar function y=f(x) is positive for any x over the interval X, then the function grows on X;
• If a similar function y=f(x) is negative, whether x is within the interval X , then the function changes to X .

In this order, in order to signify the growth of the growth and the change in the function, it is necessary:

Let's take a look at the example of the knowledge of the intervening growth and the change of the function for the explanation of the algorithm.

butt.

Know the gaps in growth and change in function.

Solution.

On the first crop it is necessary know the scope of the function. At the butt of the viraz, at the bannerman, it can turn to zero, later,.

Let's move on to the familiar function:

For the purpose of promіzhkіv zrostannya that zmenshennya funktії for a sufficient sign vyrishuєmo nerіvієmі і on the field of appointment. Be quick to use the interval method. The single root of the diary is є x = 2 and the znamennik turns to zero at x = 0. Qi points split the area of ​​the assigned interval, for some other functions, they take the sign. Significantly qi points on the number line. Pluses and minuses are mentally significant intervals, for which it is positive and negative. The arrows at the bottom schematically show the increase or change of the function on a given interval.

in such a manner, і .

At the point x=2 the function is assigned and uninterrupted, to that її should be added to the interval of growth and to the interval of decay. At the point x=0, the function is not assigned, so this point is not included in the intervals that are joking around.

We draw a graph of the function for deriving results from it.

Suggestion:

The function grows at , changing on the interval (0; 2] .

Sufficient mind the extremum of the function.

For knowing the maximum and minimum of the function, one can koristuvatisya whether one of the three is a sign of an extremum, obviously, as the function satisfies your mind. The widest and most handy are the first of them.

Persha is sufficient for Umov's extremum.

Let the function y=f(x) be differentiated in the vicinity of the point, but without interruption in the point itself.

In other words:

Algorithm of finding the point to the extremum after the first sign of the extremum of the function.

• We know the scope of the assigned function.
• We know the functions of the assigned area.
• Significantly zeros of the number dial, zeros of the banner of the corresponding point of the designated area, in which there are no possible extreme points, passing through qi points, it is possible to change your sign).
• Qi dots split the area designated for the function of promyzhki, for some it is better to take the sign. We can see the signs of a similar skin interval (for example, calculating the value of a similar function in any point of a well-taken interval).
• We select points, in which the function is uninterrupted and, passing through yaks, it changes the sign - stench extremum points.

Too rich words, more beautifully looked at the kіlka applied the significant points to the extremum and the extremums of the function for the help of the first mind enough extremum of the function.

butt.

Know extremum functions.

Solution.

The area of ​​function is all impersonal day numbers, Krim x = 2 .

We know I’ll go:

The zeros of the numerator є points x = -1 і x = 5 znamennik turns to zero at x = 2 . Significant number of points on the numerical axis

Signs of a similar skin interval are visible, with which the value of a similar skin interval is calculated, for example, at points x=-2, x=0, x=3 and x=6.

Also, on the interval it is positive (a plus sign is put on the little one above the cim interval). Similarly

We put a minus over another interval, a minus over a third interval, a plus over a quarter.

Lost to choose points, for which the function is uninterrupted and її pokhіdna change sign. Tse i є extremum points.

At the point x=-1 the function is uninterrupted and gradually changes the sign from plus to minus, then, after the first sign to the extremum, x=-1 is the point to the maximum, the second is the maximum of the function .

At the point x=5 the function is uninterrupted and gradually changes the sign of the minus to a plus, then, x=-1 is the point of the minimum, which means the minimum of the function .

Graphic illustrations.

Suggestion:

REVERSE RESPECT: the first sign is sufficient for the extremum, it does not affect the differential function of the point itself.

butt.

Find extremum points and extrema functions .

Solution.

The scope of the function is all impersonal real numbers. The function itself can be written in the view:

We know the following functions:

At the point x=0 is not possible, the shards of the values ​​of one-sided inters are not allowed to reach zero when the argument is exaggerated:

At the same hour, the output function is uninterrupted at the point x=0 (div. split follow-up of the function for continuity):

We know the meaning of the argument, under which it is worth turning to zero:

Significantly all points on the number line and significantly lower sign on the skin intervals. For which it is possible to calculate the value of the relative at certain points of the skin interval, for example, with x=-6, x=-4, x=-1, x=1, x=4, x=6.

Tobto,

In this order, after the first sign of the extremum, the points of the minimum , points to the maximum є .

Calculation of the minimum functions

Calculating the maxima of the function

Graphic illustrations.

Suggestion:

.

Another sign of the extremum of the function.

Like a bachete, for a sign of an extremum of a function, it will require a similar one, at least to a different order in points.

The first sufficient sign of the extremum is formulated with the improvement of the change of the sign of the first good hour of the transition through the critical point. About another sign of the extremum, see below in § 6.4.

Theorem (the first sign of the extremum) : YakschoX 0 - Critical point of the functiony=f(x) and in the real vicinity of the pointX 0 , passing through it zlіva to the right, pokhіdna change the sign to the prolongation, thenX 0 є extremum point. Moreover, as the sign of the opposite is changed from “+” to “-”, thenX 0 is the maximum point, andf(x 0 ) is the maximum of the function, and it is similar to change the sign from “-” to “+”, thenX 0 is the minimum point, andf(x 0 ) - Minimum function.

Looking extreme to wear local(Misceviy) character and susceptibility of a small outskirts of the critical point.

Points of extremum and points of expansion divide the area of ​​the assigned function of the interval of monotonicity.

Example 6.3. For example 6.1. we knew the critical points X 1 =0 і X 2 =2.

Of course, what is true at these points is the function y=2x 3 -6x 2 +1 may extremum. Imagine in її pokhіdnu
meaning X, taken zliva and right-handed at the point X 1 =0 to dosit near the outskirts, for example, x=-1і x = 1. taken. Oskіlki pokhіdna change the sign from “+” to “-”, then X 1 =0 - point to the maximum, and the maximum of the function
. Now we take two values ​​x = 1 i x = 3 from the vicinity of another critical point X 2 =2 . It has already been shown that
, a
. Oskіlki pokhіdna change the sign from “-” to “+”, then X 2 =2 - The minimum point. And at least functions
.

To know the most and least value of the function without interruption to the wind
it is necessary to calculate the її values ​​at all critical points and kintsyah vіrіzka, that bv choose the most and least.

6.3. Signs of swelling and shrinkage of the graph of the function. Kink points

The graph of the differentiated function is calledopuklimat the interval, like the wines of the roztashovaniya lower for whether it was your dotichnu at that interval;bend down (dip down)yakscho vіn raztashovaniya vshee be-yakої dotichї on the interval.

6.3.1. Necessary and sufficient signs of swelling and shrinkage of the graphics

a) Required signs

What is the function scheduley=f(x) tumor on the interval(a, b) , then the friend is good
at what interval; as a scheduleintimidation on the(a, b) , then
on the(a, b) .

P st schedule function y=f(x) tumor (a, b) (Fig.6.3a). Yakshcho dotichna kovzaє vzdovzh swollen crooked zlіva to the right, її kut change badly (
), at the same time, the final coefficient of dot is changing, which means that the first time is changing
on the (a, b) . Ale, however, is similar to the first one, as it is similar to the recessive function, but it can be negative, tobto
on the (a, b) .

What is the function schedule intimidation on the (a, b) , That, mirkuyuchi similarly, Bachimo, that when forging a dotic vzdovzh curve (Fig. 6.3b) cut a sickly dotichnoi growth (
); And even if it looks like a growing function, it can be positive, so
on the (a, b) .

b ) Sufficient signs

Like for functiony=f(x) all points will have the same interval
, then the graph of the functionintimidation at what interval, but how
, thentumor .

"Rule Doshu" : In order to remember some sign of another pokhіdnoї pov'yazuvati z swollen, and which one from the curved arc of the graph, it is recommended to remember: plus water in crooked lunates, "minus water" - in bulging lunares (Fig. 6.4).

Krapka graphics uninterrupted function, in which the bulge changes to the bulge of the chi navpak, is calledkink point .

Theorem (sufficient for the sign of the inflection point).

Yakscho at the point function
dvіchі differentiated that friend is similar in tsіy point to zero or not, and even when passing through the point good friend
change the sign, then the dot є point of inflection. Kink point coordinates
.

Points, for some friend, it is possible to turn to zero or not, are called critical points of a different kind.

Example 6.4. Know the points of inflection and signify the intervals of swelling and indentation of the curve
(Curve Gaus).

R solution. We know pershu that friend pokhіdnі:
,. A friend is good for you . Equal to zero and virishima otrimane equal
, de
also
, stars
,
- Critical points of a different kind. Reversing the change of the sign of another good hour for crossing the critical point
. Yakscho
for example,
, then
, but
for example,
, then
Tobto friend change the sign. Otzhe,
- abscissa of the kink point, її coordinates
. Through parity functions
, mottled
, symmetric point
, tezh will be a point of inflection.

Theorem (the first is sufficient for Umov's extremum). Let the function be uninterrupted at the point, but if the hour passes through the point, the sign changes. Todi - extremum point: the maximum, which means the sign changes from "+" to "-", and to the minimum, which means "-" to "+".

Bringing. Come on with i for .

For Lagrange's theorem , de .Todі yakshcho, then; to that , otzhe, , or . Well, then; to that , otzhe, or .

Otzhe brought, scho at any points nearby, tobto. is the maximum point of the function.

The proof of the minimum point theorem is carried out in a similar way. Theorem finished.

As soon as the hour passes through the point, it doesn’t change the sign, then the point is not extremum.

Theorem (a friend is sufficient for Umov's extremum). Let the point have a similar function, which is binary differentiating, getting 0 (), and the other one is similar in the current point as zero () and without interruption in the active neighborhood of the point. Todi - extremum point; at which point is the minimum, and at which point is the maximum.

Algorithm for extremum function recognition after the first sufficient reason to solve the extremum.

1. Know the trick.

2. Designate the critical points of the function.

3. Follow the sign of the left-handed and right-handed in the skin critical point and the growth of the visnovo about the manifestation of extremes.

4. Know the extreme values ​​of the function.

Algorithm for extremum function recognition for the help of another sufficient reason to eliminate the extremum.

1. Know the trick.

2. Know friend pokhіdnu.

3. Know tі points, u yakikh.

4. At these points, assign a sign.

5. Zrobiti vysnovok about the nature of the extremums.

6. Know the extreme values ​​of the function.

butt. Look at . We know . Daly, at i for . Dolіdzhuєmo critical points for the help of the first sufficient mind extremum. Maybe, what for i at , i at . At the points i it is better to change their sign: at "+" to "-" and at "-" to "+". Tse means that the point function has a maximum, and the point has a minimum; . For equalization, we must reach the critical point after the help of another sufficient mind and extremum. Let's know a friend will die. May: , and tse means that the point has a maximum function, and the point has a minimum.

Understanding the asymptotics of the graph of a function. Horizontal, weak and vertical asymptotics. apply.

Appointment. p align="justify"> The asymptote of the graph of the function is called a straight line, which allows you to move from the point to the center of the straight line to zero when the point of the graph is not far away from the cob of coordinates.

Distinguish vertical (Fig. 6.6 a), horizontal (Fig. 6.6 b) and sway (Fig. 6.6 c) asymptotes.

On fig. 6.6a is shown vertical asymptote.

In Figure 6.6b - horizontal asymptote.

On fig. 6.6v - asymptote.

Theorem 1. At the points of the vertical asymptotes (for example, ) the function knows the difference, between the lines and the right-handed way of the points are:

Theorem 2. Let the function be appointed to finish the great and establish final borders

І .

Then it is straight, a shabby asymptote of the graph of the function.

Theorem 3. Let the function be appointed for dosit great and іsnuє between functions. Then the straight line is the horizontal asymptote of the graph of the function.

Horizontal asymptote є we call it a bad asymptote, if . To that, although in a straight line the curve has a horizontal asymptote, then in that straight line there is no bad and bad luck.

butt. Know the asymptotics of the graph of the function.

Solution. At the point, the function is not assigned, we know between the functions left-handed and right-handed at the point:

; .

Also, is a vertical asymptote.

The main scheme for the follow-up of functions and encouragement of their schedules. butt.

General scheme of the follow-up function that prompt її graphic.

1. Know the target area.

2. Follow up the function for parity - unparity.

3. Know the vertical asymptotics of the point of expansion (like є).

4. Follow up the behavior of the function in inconsistency; know the horizontal and sickly asymptotes (like є).

5. Find extrema and intervals of monotonicity of the function.

6. Find the points of the line of the graph with the coordinate axes i, as it is necessary for a schematic diagram, to know the additional points.

7. Schematically call the schedule.

Detailed scheme follow-up functions that encourage graphics .

1. Know the destination area .

a. Yakshcho є znamennik, vin is guilty of zratatisya in 0.

b. The sub-root of the root of the paired stage can be non-negative (more than chi is equal to zero).

c. Sublogarithmic virase can be positive.

2. Follow the function for parity - unparity.

a. Yakscho , then the function is paired.

b. Yakshcho , then the function is unpaired.

c. Yakshcho not vikonano no, no , then is the function of the global view.

3. Know the vertical asymptotics of the point of expansion (like є).

a. The vertical asymptote may be less pronounced on the inter-regions of the assigned function.

b. Yakscho (or ), then the graph asymptote is vertical.

4. Follow the behavior of the function in inconsistency; know the horizontal and sickly asymptotes (like є).

a. Yakscho, then the asymptote of the graph is horizontal.

b. Yakshcho i then the straight line is a frail asymptote of the graph.

c. As for the boundaries, designated in paragraphs a, b, it is only possible with one-sided pragnennі to inconsistency (or ), then the asymptotics will be taken away one-sided: left-sided if and right-sided if.

5. Find extrema and intervals of monotonicity of the function.

a. Know pokhidnu.

b. Know the critical points (ti points, de chi de nemaє).

c. On the numerical axis, designate the designated area and її critical points.

d. On the skin of the contents of the numerical intervals, mark the sign of the next.

e. According to the signs of similar researches of visnovoks about the manifestation of extremes in those types.

f. Know extreme values.

g. According to the signs of the marching growth of the whiskers about the growth and change.

6. To know the points of the line of the graph with the coordinate axes i, as it is necessary for a schematic diagram, to know the additional points.

a. Schob to know the points of the line of the graph from the vіssyu, it is necessary to separate the line. Points , de zero , will be the points of the line of the graph z vyssyu .

b. The point of the line of the graph can be seen from the top. Vaughn іsnuє, it's less like a point to enter the area of ​​the designated function.

8. Schematically call the schedule.

a. Induce the coordinate system and asymptotes.

b. Indicate extreme points.

c. Specify the break points of the graph with the coordinate axes.

d. Schematically induce the graph in such a way that, passing through the designated points and approaching the asymptotes.

butt. Follow the function and schematically induce the її schedule.

2. - the function of a wild mind.

3. Oskіlki i , then straight lines є vertical asymptotes; dots і є dotted. , when do not enter to the area of ​​assigned function