MIR - LML - Shilov G. E. - Plotting Graphs

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nOnYll j fpHblE llEKUJ,1H no MATEMATHKE

r E. lIIMJIOBKAK CTPOI1ThrPA


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LITTLE MATHEMATICS LIBRARY

G.E.ShilovPLOTTINGGRAPHS

Translated from the RussianbyS. Sosinsky

MIR PUBLISHERSMOSCOW


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First published 1978

Ha aH2AUUCKOM f lJblKe. English translation, Mir Publishers, 1978


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The graph of the sine, wave after wave,Flows along the axis of abscissas...(FROM STUDENT LORE)

It would be hard to find a field of science or social life wheregraphs are not used. We have all often seen graphs, for example,of industrial growth or increase in labour productivity. Naturalphenomena like daily or annual fluctuations of temperature oratmospheric pressure are also easier to describe by means of graphs.The plotting of graphs of that kind presents no difficulty, providedthe appropriate tables have been compiled. But we shall bedealing here with graphs of a different kind, with graphs thatmust be plotted from given mathematical formulas. A need forsuch graphs often arises in various fields of knowledge. Thus, inanalysing the theoretical course of some physical process, ascientist obtains a formula yielding some magnitude with whichhe is concerned, for example, the amount of product obtainedrelative to time. The graph plotted from this formula will providea clear picture of the future process. Looking at it, the scientistmay possibly introduce substantial changes into the scheme ofhis experiment in order to obtain better results.In this booklet we shall consider some simple methods ofplotting graphs from given formulas.Let us draw two mutually perpendicular lines on a plane, onehorizontal and the other vertical, denoting their intersection by O.

II We shall call the horizontal line the axis of abscissas and thevertical the axis of ordinates. Each axis will be divided by thepoint 0 into two semi-axes, positive and negative, the right halfof the axis of abscissas and the upper half of the axis ofordinates being taken as positive, while the left half of the axisof abscissas and the lower half of the axis of ordinates beingtaken as negative. Let us mark the positive semi-axes by arrows.The position of any point M on the plane can now be determinedby a pair of numbers. To find them we drop perpendicularsfrom M to each of the axes; these perpendiculars interceptsegments OA and OB (Fig. 1) on the axes. The length of segment OA,taken with a plus sign + when A is on the positive semi-axis andwith a minus sign - when it is on the negative semi-axis, we call

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the abscissa of point M and denote by x. Similarly, the lengthof segment OB (with the same sign rule) we call the ordinateof point M and denote by y. The two numbers x and yarecalled the coordinates of point M. Every point on the plane hascertain coordinates. The points on the axis of abscissas have zeroordinates and the points on the axis of ordinates have zero abscissas.The origin of the coordinates 0 (the point of intersection of the

Axyo

B -------- IMIIII

Fig. 1axes) has both coordinates equal to zero. Conversely, when we aregiven any two numbers x and y (negative or positive), we canalways plot a point M with abscissa x and ordinate y; to do thatwe measure off a segment OA = x on the axis of abscissas anddraw a perpendicular AM = y at point A (bearing in mind thesign); M will be the point we want.

Suppose we are given a formula for which we are to plotthe graph. The formula must indicate what operations must becarried out on the independent variable (denoted by x) in orderto obtain the value we need (denoted by y). For example, theformula

y = 1 + x2indicates that the value of y can be obtained by squaring theindependent variable x, adding it to unity and dividing unity bythe result. If x assumes some numerical value .xo, then inaccordance with our formula y will assume some numerical value Yo.The numbers Xo and Yo will determine a certain point M 0in the plane of our picture. Instead of Xo we can take "someother number Xl and use the formula to calculate a new value YI;the pair of numbers (Xl ' YI) will define a new point M 1 on theplane. The locus of all points whose ordinates are related to theirabscissas by the given formula is called the graph of this formula.The set of points on a graph, generally speaking, is infinite,

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and we cannot hope in fact to plot them all without exceptionaccording to the given rule. But we can manage without it.In most cases it is enough to know a small number of pointsin order to be able to judge the general form of the graph..,

o oo o oFig. 2

The point-by-point plotting of a graph consists simply inplotting a certain number of points of the graph and then joiningthem (as far as possible) by a smooth curve.As an example, let us consider the graph of the function

y = 1+ x 2Let us compile the following table:

(1)

x 0 1 2 3 -1 -2 -3y

In the first line we have written down values for x = 0, 1, 2,3, -1 , - 2, - 3. We usually take integral numbers for x becausethey are easier to operate with. In the second line we havey

o- - - - - - 1 ~ - ~ I _ _ _ - _ _ _ _ _ f - - ~ - - _ + - - _ _ + . - - _ _ + ~ XFig. 3

written the corresponding values for y obtained from formula (1).Let us plot the corresponding points on the plane (Fig. 2).Joining them by a smooth line, we obtain the graph (Fig. 3).7


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(2)y = (3x2 - 1)2

The point-by-point plotting, as we see, is extremely simpleand requires no 'theory' But, perhaps just because of that, we canmake bad mistakes by- blindly following it.Let us plot the curve given by the equation1

using this rule. y

o o

Fig. 4The table of values for x and y corresponding to this equationis as follows:

x o 2 3 -1 -2 -3y 1/6 76The corresponding points on the plane are shown in Fig. 4.The outline looks very like the previous one; joining the pointsby a smooth line we obtain a graph (Fig. 5). Now it seems

oFig. 5

we can put down our pens and take a deep breath; we havemastered the art of plotting graphs! But just in case, as a check,let us calculate y for some intermediate value of x, say x = 0.5.We get a surprising result: for x = 0.5, y = 16. This sharply disagreeswith our picture. And there is no guarantee that in calculating8


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y for other intermediate values of x - and there is an in'initenumber of such values - we would not obtain even more strikingincongruities. Apparently the point-by-point plotting of a graphitself is not well-founded.** *Now we shall consider another method of plotting graphs, morereliable since it protects us from those unexpected things we havejust come across. This method - let us call it 'plotting by operations' - consists in carrying out directly on the graph all theoperations which are written down in the given formula: addition,subtraction, multiplication, division, etc.Let us consider some of the simplest examples. We plot thegraph corresponding to the equation

y = x (3)This equation shows that all the points of the sought line on

- - - - - - # - - - - - ~ X

Fig. 6

- - ~ - + - - - - # - - - + - - - t - - ~ X

y= kxwith some coefficient k is obtained from the previous one bymultiplying each ordinate by the same number k. Suppose, forexample, k == 2; each ordinate of the previous graph must bedoubled. and .. as a result, we obtain a straight line more steeply

Fig. 7the graph have equal abscissas and ordinates. The locus of pointshaving an ordinate equal to an abscissa is the bisector of theangle formed by the positive semi-axes and the angle formedby the negative semi-axes (Fig. 6). The graph corresponding to theequation

2-19 9


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rising upwards (Fig. 7). Each step to the right along the x..axiscorresponds to two steps along the y-axis. Incidentally, it is veryconvenient to plot such graphs on squared or graph paper. In thegeneral case we will also get a straight line in the equationy = kx. If k > 0, the line, with each step to the right, will moveup k steps along the y-axis. If k < 0, the line will slope downinstead.

- - - - # - ~ - - - ~ x

Fig. 8Now let us consider a somewhat more complicated formula

y = kx + b (4)To plot a corresponding graph, we must add one and the

same number b to each ordinate of the already known line y = kx.As a result, the line y = kx will move up, as a whole, by bunits if b > 0; if b < 0, the original line will, of course,movedown instead. We will thus obtain a straight line parallel to theinitial one, but no longer passing through the origin of coordinatesand intercepting the segment b on the axis of ordinates (Fig. 8).Therefore, the graph of any polynomial of degree one in xis a straight line which can be plotted according to the aboverules.Now let us consider the graph of polynomials of degree two.Let us consider the formula

y = x 2 (5)It can be represented In the form

Y =YiwhereYt = x

In other words, we will obtain the sought graph by squaring10


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each ordinate of the known line y = x. Let us see what we get in thatway.Since 02 = 0, 12 = 1, ( - 1)2 = 1, we have obtained three basicpoints A, Band C (Fig. 9). When x > 1, X l > x; therefore, tothe right of point B, the graph will lie above the bisector ofthe quadrant angle (Fig. 10).When 0 < x < 1, 0 < x 2 < x; therefore,between points A and B the graph will lie below the bisector.y

Co 08A x-1 0

Fig. 9Moreover, we claim that when we approach point A, the graphwill be contained in any angle bounded from above by the straightline y = kx (with an arbitrarily small k), and from below by thex-axis; indeed, the inequality

X l < kxis satisfied only if x < k. This fact means that our curve istangent to the axis of abscissas at point 0 (Fig. 11). Now let

B//// A XX0 0

Fig. 10 Fig. 112* 11

y


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us move along the x-axis to the left of point O. We know thatthe numbers - a and + a give the same square (+ a2 ). Thusthe ordinate of our curve for x = - a will be the same as forx = +a. Geometrically this means that the graph of the curvein the left half-plane will be a mirror image of the alreadyplotted graph in the right half-plane with respect to the y-axis.We have obtained a curve called a parabola (Fig. 12).

o- - ~ ~ - - - ~ ~ x- - - - + - - - - - . : . ~ - - + _ _ - ~ x

Fig. 12 Fig. 13Now by the same method we can plot a more complicatedcurve (6)and an even more complicated one

y == ax? + h (7)The first is obtained by multiplying all the ordinates of parabola(5), which we will call the standard parabola, by the number a.For a > 1 we get a similar curve but more steeply risingupwards (Fig. (3). For 0 < a < 1 the curve will be steeper (Fig. (4),and for a < 0 its branches will turn upside down (Fig. 15). Curve (7)is obtained from curve (6) by moving it up by a distance hif h > 0 (Fig. 16). But if h < 0, we will have to move the curvedown instead (Fig. 17). All these curves are also called parabolas.

Now let us consider a somewhat more complicated exampleof plotting a graph by multiplication. Suppose we are to plotthe graph according to the equation

y == x(x - I )(x - 2)(x - 3) (8)12


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y

Fig. 14

y

Fig. 15

Fig. 1613


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Here we are given the product of four factors. Let us plotthe graph of each of them separately: they are all straight lines,parallel to the bisector of the quadrant angle and interceptingthe y-axis at points 0, -1 , -2 , -3 , respectively (Fig. 18).At points 0, 1, 2, 3, on the x-axis, our curve will have a zeroordinate, since a product is equal to zero if at least one ofthe factors is zero. At other points the product will be non-zero

Fig. 17and will have a sign which can easily be found from the signsof the factors. Thus, to the right of point 3 all the factors arepositive, therefore, so is their product. Between points 2 and 3 onefactor is negative, therefore, so is the product. Between points 1and 2 there are two negative factors, so the product is positive,etc. We obtain the following distribution of signs of the product(Fig. 19). To the right of point 3 all the factors increase with

Fig. 1814


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x, therefore, the product will also increase and very rapidly. Tothe left of point 0 all the factors increase in the negativedirection, so the product (which is positive) will also rapidly increase.Now it is easy to sketch the general form of the graph(Fig. 20).So far we have used the operations of addition and multiplication. Now let us consider division. Let us plot the curve1Y = t + x2

y

+ + + x0 2 3

(9)

Fig. 19To do that we separately plot the graphs of the numerator andthe denominator. The graph of the numerator

Yt = 1is a straight line parallel to the x-axis and passing through unity.The graph of the denominator

Y2 == x 2 + 1y

Fig. 2015


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is a standard parabola, moved upwards by unity. Both of thesegraphs are shown in Fig. 21.Now we will carry out the division of each ordinate of thenumerator by the corresponding (i. e. taken for the same x)ordinate of the denominator. When x == 0, we see that Yl == Y2 == 1,which yields y == 1. When x i= 0, the numerator is less than thedenominator, so that the quotient is less than unity. Since thenumerator and the denominator are positive everywhere, thequotient is positive, and, therefore, the graph is contained in thestrip bounded by the x-axis and the line y == 1. When xinfinitely increases, the denominator also grows infinitely, whereasthe numerator remains constant; therefore, the quotient tends tozero. All this gives the following graph of the quotient (Fig. 22).We have obtained the same picture as the one plotted previouslyby the point-by-point plotting of a graph (p. 7}.When carrying out division on the graph, particular attentionshould be paid to those values for x at which the denominatorbecomes zero. If the numerator is non-zero at this point, thequotient becomes infinite. For example, let us plot the curve

1y == ~ (10)Here, the graphs of the numerator and the denominator are alreadyknown (Fig. 23). For x = 1 we have Yl == Y2 == 1, which yieldsY == 1. When x > 1, the numerator is less than the denominatorand the quotient is less than unity as in the previous example.When x increases infinitely, the quotient tends to zero and weobtain the part of the graph which corresponds to the valuesx > 1 (Fig. 24).Let us consider now the values for x betweennd 1. When xmoves from 1 to 0, the denominator tends to zero, while thenumerator remains equal to unity. Therefore, the quotient increasesinfinitely and we obtain the branch moving up to infinity(Fig. 25). When x < 0, the denominator as well as the wholefraction become negative. The general form of the graph is presentedin Fig. 26. Now we can already start to plot the graphdiscussed on page 8:

1Y == (3x2 _ 1)2 (11)Let us first plot the graph of the denominator. The curveY1 == 3x2 is a 'triple' standard parabola (Fig. 27). Subtracting unitywe move the graph down by unity (Fig. 28). The curve intersectsthe x-axis at two points which will be easily found by setting16


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Numerator

oFig. 21

oFig. 22

y

Numerator- - - - - - - # - - - I - - - - - - ~ x

Fig. 2317


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3x2 - 1 equal to zero:Xl.2 = M 0.577...

Now let us square the obtained graph. At points Xl and X2the ordinates remain equal to zero. All the other ordinates willbe positive, so that the curve will lie above the x-axis. At thepoint x = 0 the ordinate will be equal to ( - 1)2 = 1 and thisy

- - - - I - - - - + - - - - - = l ~ xNumerator

Fig. 24

y

Fig. 25will be the greatest ordinate of the part of the graph betweenXl and X2. Outside this part the curve will steeply rise upwardson both sides (Fig. 29).

Numerator---+-----Jlr---

o

Numerator

~ - - - - + - - I - - - - - + - - - - D ~ + - - - + - - - - i f - - - - - - - : ~X..

Fig. 261.8


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y

y

oFig. 27

- - - - + - - - ~ - - + - - - ~ x

Fig. 28

y y

+ - - . J U L - - + - - - . . J I I I ~ + - ~ X-1

IIIIIII Numerator~ ~ - + - - -I Denominator

oFig. 29 Fig. 30

19


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The graph of the denominator has been plotted. In the samedrawing the dotted line shows the graph of the numerator Y3 = 1.Now we only have to divide the numerator by the denominator.Since both of them have the same sign everywhere, the quotientwill be positive, and the whole graph will lie above the x-axis.When x = 0, the numerator and the denominator are equal, andtheir quotient is equal to unity. Let us move along the x-axis

y

~ooceoeI IQ

Numerator

Fig. 31to the right from point O. The numerator remains equal to unityand the denominator decreases; therefore, the quotient increasesbeginning with unity. When we reach the value X2 = 0.577...the denominator will become equal to zero. This means that thequotient will become infinite by this moment (Fig. 30). Beyondthe point X2 the denominator will quickly move in the oppositedirection from the zero value to unity and will then infinitelyincrease. On the contrary, the quotient will return from infinityto unity intersecting the straight line y" = 1 at the same point as Y3and further will indefinitely approach zero (Fig. 31).

The same picture will be obtained to the left of the axis ofordinates (Fig. 32).We have marked on this graph the points corresponding tothe integral values for x = 0, 1, 2, 3, -1 , -2 , - 3. These arethe same points which have been taken in the point-by-pointplotting of a graph on page 8. But the actual form of the graph20


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y

Fig. 32

Fig. 33

- - - - 1 - - I S - - - + - + - - + - - - - . . . . . . , f - - - + - - - + - + - - + - - - - I t - - - + - - - + - + - - + - - t - - ~ x

Fig. 3421


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radically differs from the one proposed in Fig. 5. We see thatactually instead of smoothly decreasing from the value of 1 (forx = 0) to the value of 1/4 (for x = 1) and further, the curvemoves upwards to infinity. Here we can see the point withcoordinates x = 1/2, y = 16 which had no place on the previousincorrect graph but which gets in quite nicely on the new correct one.** *Summarizing the general rules which should be applied to plotgraphs 'by operations', we can say:(a) All the operations contained in the given formula mustbe carried out with the graphs going from simple ones to morecomplicated.(b) When multiplying graphs pay attention to the points wherefactors (at least one of them) become zero; remember the rule of

signs between those points.(c) When dividing graphs pay attention to the points wheredenominator vanishes. If at those points numerator is non-zero,the branches of the curve will move to infinity - up or downdepending on the signs of the numerator and the denominator.(d) Pay attention to the behaviour of the curve when xmoves indefinitely to the right (to + (0) or to the left (to - oo].Here we have only described the simplest operations whichmay be carried out with graphs. To be more precise, we startedout with the simplest equation y = x and applied further the fouroperations of arithmetic: addition, subtraction, multiplication, anddivision. To these operations we could easily add the algebraicoperation - extracting roots.But more complicated operations - trigonometric and logarithmicones - can also be carried out with graphs. One only needs toknow the graphs of the simplest equations y = sin x (Fig. 33)and y = logx (Fig. 34). Using the methods desribed above we canplot graphs of any equations involving the symbols sin, log,and algebraic and arithmetic operations.It is very useful to learn how to plot all kinds of graphs.However, by using the methods indicated above we will not beable to answer many natural questions arising when one considerssome graphs. For example, we see on a certain graph that thecurve rises to some value Yo, then begins to move down; we sayin that case that it reaches its maximum value Yo at point Xo.We may not be able to find the exact value of Xo with ourlimited scope of methods. Other questions arise, for example, atwhat angle the curve intersects the x- or the y-axis, whether it22


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is convex upward or downward. Our methods cannot give exactanswers to all of those questions. Here, a considerably moreserious knowledge of mathematical technique is required. Themethods which allow the study of the above properties of graphsare contained in the branch of mathematics known as the differentialcalculus.** *

In conclusion solve some problems on graphs.Plot the graphs of the following equations:1. y = x 2 + X + 1. 3. y = x 2 (x - 1).2. y = x(x 2 - 1).

x5. Y=--l.x-

4. y = x(x - 1)2

x 1Hint. Single out the integral part: - - -= 1 + - - .x-I x-Ix 26. y= --1.x -

Hint. Single out the integral part.x 37. y = - - 1 - .' x -

Hint. Single out the integral part.8. y = ~ .

Hint. The square root of negative numbers does not exist Inthe real part.9 . y = ~

How to prove that this curve is a circle?Hint. Recall the definition of a circle and the Pythagoreantheorem.

1ft y = ~ .23


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Prove that the upper branch of the curve indefinitely approachesthe bisector of the quadrant angle when x ~ 00.

Hint. Vx2 + 1 - x == 1~ + x 11. y= xVx( l - x ) .

12. y= X 2 ~ .13. y= 1 ~ .2 V1 - x 214. Y == x 2/3 (1 - x)2/3 .


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Answers to.Problems

y

2To Problem

~ - + - - - + - - - + - - - ~ x-2 -1

To Problem 2

y

To Problem 3 To Problem 4

25


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y IItIIIIII- ~ - - - - - - - - - -I

I

26

To Problem 5

To Problem 6


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y 1\)1II II JI II I: I .....I 1"1-I ':I It\ft '/tI I :a.-,I)'

/ II I/ II III

ilIIIII

y

To Problem 7

To Problem 8


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- - + - - - - + - - - ~ - ~ x

To Problem 9

- - - - - - ~ - - - - - ~ x

To Problem to

y

o-.... : : - - - - - - - + - - - ......xTo Problem 11

28


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To Problem 12

y1

o. . . . I K - - - - - - - + - - - - - . . a . . . . - ~ x-1

To Problem 13

oTo Problem 1429


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Recommended Literature

The Encyclopedia of Elementary Mathematics, Book 3, Gostekhizdat(19.52), the article by V. L. Goncharov "Elementary Functions", Chaps.1 and 2.


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TO THE READER

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MIR - LML - Shilov G. E. - Plotting Graphs