5 definition of a fractional linear function. Graphing functions is one of the most interesting topics in school mathematics.

Literacy - urgent need for many people. In our country it is held annually total dictation. Important officials and people of other professions in dozens of cities are diligently thinking about how to correctly write this or that word, where to put a comma, a dash, etc. There is something to think about, because Russian is not such a simple language. And it is impossible to write in it accurately without knowing the rules.

The issue of spelling is acute, primarily for schoolchildren. After all, dictations for them are not held once a year. They write testing work almost weekly different subjects. Therefore, they should know not only the rules given in this article, but also others. For example, being able to write correctly English suffixes adjectives. Or know when some famous figure was born and died...

Suffixes -chiv, -liv, -iv, -ev. Exception words

They say that you need to get into the water to learn to swim, and to learn to write without mistakes, you need to train and develop the skill of writing correctly.

The choice of vowels in a given morpheme depends on several reasons: stress, hardness or softness of the preceding sound, meaning. And even the origin of Stress and literate writing are related to each other due to the fact that only stressed vowels are clearly audible in the flow of our speech, with unstressed letters more difficult. If you don't know the rules, mistakes will appear.

Remember! If you are in doubt which adjective suffixes to write, put emphasis: -iv-- if it’s shock, but -ev-- if e is unstressed (playful, false, but key); -chiv-, -liv- are derived from -iv, so they write and (assiduous, noisy). Exception: holy fool, merciful.

Now check yourself: zero... howl, caring... howl, playful... howl, merciful... howl, bo... howl, arrogant... howl, trust... howl. (Answers: e, and, and, and, e, and, and).

Suffixes -ov, -ovat and -ev, -evat

Anton Pavlovich Chekhov did not like ugly, dissonant sounds and in every possible way avoided the use of hissing and whistling sounds. Our spelling cannot do without them.

Over the spelling of suffixes - ov -, - ev- after the unloved classics, w, w, c are often thought about. The letters o and e sound alike are confusing. It's simple: O- stressed, and without stress - e(kumac, penny, plush).

What if the stress falls on the root or prefix? In this case, adjective suffixes become dangerous for an illiterate person. Here you need to remember the rules! Last letter Basics: Should you write suffixes? -ov-, -ovat-, -ovit-(rough). And if the consonant is soft, hissing or c, then -ev-, -evat-, -evit- (key, glossy, apple, blind).

Now let’s check the knowledge we’ve gained: gifted...twisted, demanding, watchman...howl, half...howl, blue...wavy, canvas...well done...well done. (Answers: o, o, e, e, e, o, e).

Suffixes -onk and -enk

It is impossible not to be surprised by the possibilities of the Russian language. In order for a word to change its meaning, it is enough to replace the suffix in it. And now the fat one turns into a plump one, and the thin one into a thin one. Expresses a completely different attitude towards the subject of speech. The adjective suffixes -onk- and -enk- give words shades of sympathy, endearment, irony, disdain, as well as real diminution. Only they have the ability to use these formative means.

Adjectives with diminutive formed by means of a suffix -enk and - onk(blue, plump).

After the letters g, k, x, and are possible -onk and -enk(light and light).

Test yourself! Bad...cue, round...cue, beautiful...cue, good...cue (oh, e, e, e).

Showing your attitude to what is being said helps and -okhonk, -okhonk, -usenk, -usenk(small, tiny, thin).

Suffixes -ensk, -insk. Exceptions

Adjectives are formed not only from common nouns, but also from proper nouns and geographical names. Morphemes -insk- and -ensk- are used to form such words. Unstressed suffixes of adjectives sound unclear. Such speech situations when it is necessary to both write and use a word in oral speech. Wrong educated form hurts the ear. If you don't want to make mistakes, remember the rule.

Suffix -insk- serves to form nouns with final -a, -я, -и. (Yalta - Yalta, Mytishchi - Mytishchi, Ekaterina - Catherine). In other cases the suffix is ​​used -ensk-(Grozny - Grozny, Grodno - Grodno).

The Russian language is beautiful. And he does not tolerate cacophony, so there are exceptions to the rules that are worth learning. These are Presnensky (from Presnya), Penza (from Penza), Kolomna (from Kolomna), etc.

It is impossible, and it is not necessary, to remember exceptions to all the rules. If you have any difficulties, you should consult dictionaries and reference books.

Spelling of adjectives formed from the nouns na-shka, the words “wind” and “oil”

Nouns with finite -shka(frog, cat) form adjectives, about the spelling of which you need to know the following: both “frog” and “frog” are written, cat - cat. Both spellings are correct. Only the emphasis is different.

The spelling of some adjectives is related to their lexical meaning. From the words “wind” and “oil” you can form word forms that have different suffixes adjectives. Examples of such words: anemone - windy, oily - oily. There are no mistakes here.

Worth remembering! 1. Windy (with the meaning of “wind” in literal and figurative meaning. The wind on the street (direct) and the wind in the head (trans.) 2. Windy (with the meaning “set in motion by the force of the wind”) 3. Chicken pox ( from spoken word"chickenpox"). 4. Oily (consists of any oil). 5. Oiled (greased, soaked, stained with oil). 6. Oily eyes ( figurative meaning). 7. Maslenitsa - Maslenitsa week.

Suffixes -sk and -k. Exceptions

“It’s not a shame not to know, it’s a shame not to learn,” says folk wisdom. And it’s worth constantly expanding your knowledge. Spelling adjective suffixes is often difficult. In the words "sharp" and "sailor" the end of the word sounds the same. How to choose the correct spelling? Let's think like this. Since adjectives are formed from the stem of a noun, then we highlight this stem and see which letter is the last (weaver - weaver). If k, h, ts- feel free to substitute the suffix -To. Alternation may occur. This is normal. Are there no such letters? Create a shape. Did it work? Then feel free to write the suffix -To.

At the end of the base k, h, ts(giant - gigantic), and it’s impossible to form a short form - feel free to write the suffix -sk.

Remember: b before -sk- written only after l, and also in the words: day-day, names of months, but we write “January” without a soft sign.

Adjectives starting with -aty: suffixes -schat, -chat

Mikhail Sholokhov said that the language of the people has been accumulating treasures of thought and experience for thousands of years. Formed from nouns using the suffix - chat-, -chat- the words seemed to bring mystery and charm from time immemorial (iridescent, log, patterned). Indicate presence in large quantities of what is expressed by the base (fragmentary).

The spelling of adjective suffixes depends on the front standing letters. Alternation occurs, which changes the composition of the morpheme and its sound.

An error may appear when writing.

Dosch-at-y (from board sk/sch).

Smoke-chat (from smoke).

Brus-chat-k-a (from brus).

Tile-chat (tiles c/t).

Adjectives with such suffixes are widely used in professional and terminological vocabulary.

not formed from verbs

“The old abandoned garden was beautiful in its picturesque desolation,” we read from the classic. But in order to excel in literacy in dictation, you need to remember the rule of how to choose suffixes of adjectives and participles. The difficulty is that it is impossible to determine by ear "n" or "nn" choose.

Let's remember the rule! If you forgot, look in the reference book. Next we analyze how it was formed given word(picture-n y - base on -n, but a suffix was added -n - here is the double "n").

"Nn" it is also written when the adjective appears with the help of the suffix -onn-, -enn- (excursion-onn-y, craft-enn-y).

The letter "n" is written in adjective suffixes if in the word -in-, -an-, -yang-(os-in-y, skin-an-y).

Exceptions:

1) glass, tin, wood;

2) young, green, porky, spicy, drunk, windy, crimson.

Spelling "n", "nn" in adjective suffixes it makes you rack your brain if you don’t know the rules. Therefore, it is worth training them.

Spelling "n" and "nn" in nouns and adverbs starting with -o, formed from non-verbal adjectives. Words to remember to spell

The vocabulary wealth of our language is constantly growing. One of the ways of replenishment is the formation of new names of objects and phenomena from the names of adjectives. "N", "nn" in adjective suffixes determines the number of letters in formed nouns. Examples: living room - hotel, relative - relative, wood - woodshed, horse - cavalry.

Adjectives can also form adverbs with a final -O. Interestingly, the suffixes of adverbs and adjectives are the same. The number of "n" is also the same. You can check the spelling of “n”, “nn” in adverbs by choosing full adjective, from which the adverb is derived. Write as many of these letters as in the original adjective. Examples: courageously - courageous, magnificent - magnificent, excitedly - excited.

Remember the spelling of the following words: welcome, smart, uninvited guest, sworn brother, unprecedented, unheard of, etc.

  • § 7. Not at the beginning of the root after vowels (in words of foreign origin) both the letter e and e are written. Their choice depends on the preceding vowel.
  • § 8. Not at the beginning of the root after consonants, the letter e is written to convey the vowel e and at the same time to indicate the hardness of the preceding consonant in the following cases.
  • § 9. In other cases, the letter e is written not at the beginning of the root after the consonants.
  • § 10. In the following cases, the letter e is written to convey the combination j followed by the vowel e:
  • § 11. The letter is written:
  • § 12. The letter ы is used to convey the vowel ы and at the same time to indicate the hardness of the preceding consonant:
  • § 14. After zh, sh, h, sch the letter i is written (and s is not written), for example: fat, saffron milk cap, interpublishing, say, sew, reeds, clean, rays, shield, look.
  • § 15. After q the letter i or s is written.
  • § 17. After zh, sh, h, shch to convey the stressed vowel e, the letter e is written, for example: tin, swing, rustle, ginseng, chirp, zhe (name of the letter), on the boundary, about a candle, a soul, a sling; Zhenya, Jack, Shannon.
  • § 18. After zh, ch, sh, shch, the letter o or e is written to convey the stressed vowel o.
  • § 19. In all other cases, to convey the stressed vowel o after zh, ch, sh, sh, the letter e is written, namely:
  • § 25. The letter e is written after the letters zh, ch, sh, c only in the following special cases.
  • § 27. The separative ъ is written after the consonants before the letters i, yu, ё, e, conveying combinations [j] with vowels, in the following cases.
  • § 29. The letter ь is written to indicate the softness of a paired consonant at the end of words, for example: dove, leave, notebook, dirt, pity, seven, horse, kitchens, apple trees, embankment, beast, paint, drink, shipyard.
  • § 30. To indicate the softness of a paired consonant before consonants, the letter ь is written in the following cases.
  • § 31. The letter ь is written (regardless of pronunciation) in the following grammatical forms:
  • § 32. After zh, sh, ch, shch, the letter b is written according to tradition in the following grammatical forms:
  • § 33. General rule. The writing of letters in place of unstressed vowels is established by checking other words and forms, where in that
  • § 35. There are roots in which writing letters in place of unstressed vowels does not correspond to the general rule, but is subject to tradition. These include the following roots with alternating vowels.
  • § 43. Suffixes with unverifiable unstressed vowels.
  • § 46. -Ev-, -iv-, -liv-, -chiv- (in adjectives). It is necessary to distinguish between adjectives with the suffixes -ev-, on the one hand, and -iv-, -liv-, -chiv, on the other.
  • § 51. -Ink-, -enk-, -ank- (-Yank-). It is necessary to distinguish between nouns in -Inca and in -enka (with unstressed vowels before n).
  • § 55. -Insk-, -ensk-. It is necessary to distinguish between the suffixes of the adjectives -insk- and -ensk- (with the letters i and e in place of the unstressed vowel).
  • § 58. -Usch- (-yush-), -ash- (-box-). In the suffixes of active present participles the same vowels are written as in the 3rd person plural forms. Parts of the same verbs (see § 74):
  • § 64. Unstressed fluent vowels are represented by the letters e, o or and according to the following rules (below, after each example, a form or word is given in parentheses where there is no fluent vowel).
  • § 66. Instead of the connecting vowels o and e in some categories of compound words, vowels are written that coincide with the case and other endings of words, the stems of which are contained in the previous part of the word:
  • § 69. The following endings contain vowels that are not checked by the stressed position.
  • § 70. Nouns with some suffixes have features in the writing of unstressed endings.
  • § 71. Case forms of nouns in -i, -i, -i.
  • § 76. The writing of letters in place of unstressed vowels before the suffix - the infinitive is determined by the following rule.
  • § 78. There are special constructions with particles not and neither.
  • § 86. Letter combinations tch, dch (including stch, zdch) are written at the junctions of significant parts of a word if the previous part of the word ends in t or d, and the subsequent one begins with h, for example:
  • § 88. The letter combinations сч, зч, zhch, shch are written at the junctions of significant parts of a word if the previous part of the word ends in s, z or zh, sh, and the subsequent one begins with h, for example:
  • § 89. The letter combinations сш, сж, зш, зж are written at the junctions of significant parts of a word if the previous part ends with a consonant s or z, and the subsequent part begins with sh or zh, for example:
  • § 95. Double n and double s are written at the junction of a generating stem and a suffix, if the stem ends and the suffix begins with the same consonant n or s:
  • § 103. Short forms of complex adjectives, the second parts of which coincide with participles in -nny, are written with n or nn depending on the meaning. Adjectives expressing characteristics
  • § 106. Double consonants are written in the roots of Russian (not borrowed) words in the following cases.
  • § 46. -Ev-, -iv-, -liv-, -chiv- (in adjectives). It is necessary to distinguish between adjectives with the suffixes -ev-, on the one hand, and -iv-, -liv-, -chiv, on the other.

    Words starting with -evy - for example, soy, queen, flannel, match, suede - contain the suffix -ev- (a spelling variation of the suffix -ov-), which is checked under the stress in words such as oak?vy, ruble. These adjectives are relative adjectives and do not have short forms.

    Words starting with -ivy are qualitative adjectives (having short forms) with suffixes -iv-: merciful; -liv-: caring, envious, restless, whimsical; -chiv-: forgetful, resourceful, changeable; Wed suffixes -iv- and -liv- under stress: lazy, funny, whimsical.

    Note. The suffix -iv- is also found in the word holy fool, formed from the obsolete yurod.

    § 47. -enk-, -onk- (in nouns). In nouns with the suffix -enk- (-onk-), the initial vowel of the suffix, always unstressed, is represented in writing by the letters e (after paired soft consonants and sibilants, as well as after vowels) and o (after paired hard consonants): daddy, daughter, little leg, darling, Petenka, Varenka, Seryozhenka, Mashenka, Zoenka; birch, baby, kitty, little teeth, Veronka.

    Exceptions: in the words zainka and good boy, as well as in the word bainki, the letter i is written in the suffix.

    Note 1. writers of the XIX V. spellings of proper names such as Marfinka, Polinka, Fedinka (with the letter i), as well as Lizanka, Lisanka (with the letter a; the latter - in folklore texts) are widely found. Such spellings, although they deviate from the modern spelling norm, are preserved in reprints of the corresponding texts.

    Note 2. In texts oriented towards folklore tradition, there is also something deviating from modern norm writing this suffix with the letter ы after a hard consonant, for example: You can hear the cry of the neighboring stripe, / The woman is there - her kerchiefs are disheveled, / We need to rock the child! (Necr.); For Nadenka the bride-to-be, there is a dowry in the closet (Rozhd.); It will roll and roll - that’s it, I think my death has come! (Shuksh.). Writing -ynk- at the end of stems (not under stress) is normal for words formed with the suffix -k(a) from nouns starting with -ynya: alms - alms, barynya - lady.

    § 48. -points-, -points-. In nouns with the suffix -ochk- (-echk-), the initial unstressed vowel of the suffix is ​​expressed by the letters o (after paired hard consonants) and e (in other cases): light bulb (from lamp), bath, vase, blouse, mommy, Ninochka, Allochka , Vovochka; auntie, Vanechka, Olechka, Raechka; time (from time), name (from name), seed, morning. The emphasis on the suffix is ​​a place in a single word.

    Nouns with the diminutive suffix -k-, formed from nouns ending in -ka, -ko, -ki, have the same orthographic appearance: detochka (from detka), distaff, papillary, little fence, sleigh (from sled); tiny, leechka, window (from window), etc. (in them the letters o and e convey a fluent vowel, see § 64).

    Note. They deviate from the modern norm of spellings such as Tetichka, Volodichka, Raichka (as well as the stylized spelling Venichka Erofeev, traditional in modern texts). Writing -ichk- at the end of stems (not under stress) is normal for words formed with the suffix -k- from nouns with a stem in -its-, -ik-, for example: staircase - staircase, mill - mill, button - button , scissors - scissors, factory - factory, Edik - Edichka.

    § 49. -achiy (-yachiy), -echy, -ichy. It is necessary to distinguish between adjectives with the stressed suffix -achiy (-yachiy) such as cat?chiy, frog?chiy, turkey?chiy, duck?chiy, pig?chiy and adjectives nachiy with the suffix -y, where the letter e conveys an unstressed fluent vowel ( see § 64), e.g.: boy?shechy (from boy), old?shechy (old woman), cuckoo (cuckoo), fro?shechy, indy?shechy.

    In the words be?lichiy, ne?rpichiy and parrot?ichy, the unstressed vowel at the beginning of the suffix -ichy is represented by the letter i.

    § 50. -ek, -ik. It is necessary to distinguish between the diminutive suffixes of masculine nouns -ek (in an unstressed position) and -ik, -chik (always unstressed). In the suffix -ek, the vowel (checked under stress in words like penyok, cottage cheese) is fluent, and in the suffixes -ik and -chik the vowel is preserved during declension. Thus, a fluent vowel is conveyed here by the letter e (see § 64), and a non-fluent vowel by the letter i, for example: ovrazhek - ravine, piece - piece, knife - knife, lamb - lamb, but table - table, kalachik - kalachik, glass - glass.

    According to the same rule, formations characteristic of colloquial speech and vernacular are written in -ik with a non-fluent vowel of the suffix and a truncated base of the producing word such as velik (bicycle), telly (TV), vidik (video recorder), shizik (schizophrenic), as well as truncated diminutive personal names like Alik, Vladik.

    Here the coefficients for X And free members in the numerator and denominator - given real numbers. Schedule fractional linear function V general case is hyperbola.

    The simplest fractional linear function y = - You-

    strikes reverse proportional dependence ; the hyperbola representing it is well known from the course high school(Fig. 5.5).

    Rice. 5.5

    Example. 5.3

    Plot a graph of a linear fractional function:

    • 1. Since this fraction does not make sense when x = 3, That domain of function X consists of two infinite intervals:
    • 3) and (3; +°°).

    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 at 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:

    • a) for x->3 right(i.e. for *>3) the value of the function increases without limit: at-> +°°: at x->3 left(i.e. at x y - Thus, the desired hyperbola approaches the straight line without limit with the equation x = 3 (bottom left And top right) and thus this straight line is vertical asymptote hyperbole;
    • b) when x ->±°° the second term decreases without limit, so the value of the function approaches the first, constant term without limit, i.e. to value y = 2. In this case, the graph of the function approaches without limit (bottom left and top right) to the straight line given by the equation y = 2; thus this line is horizontal asymptote hyperbole.

    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).

    • 3. Assuming l = 0, we find y = ~. Therefore, the desired hy-

    perbola intersects the axis Oh at the point M x = (0;-^).

    • 4. Function zero ( at= 0) will be at X= -2; therefore, this hyperbola intersects the axis Oh at point M 2 (-2; 0).
    • 5. A fraction is positive if the numerator and denominator have the same sign, and negative if they have different signs. Solving the corresponding systems of inequalities, we find that the function has two positive intervals: (-°°; -2) and (3; +°°) and one negative interval: (-2; 3).
    • 6. Representing a function as a sum of two terms (see item 2) makes it quite easy to detect two intervals of decrease: (-°°; 3) and (3; +°°).
    • 7. Obviously, this function has no extrema.
    • 8. Set Y of the values ​​of this function: (-°°; 2) and (2; +°°).
    • 9. There is also no even, odd, or periodicity. Information collected enough to schematically

    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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