Booklet shilov plotting-graphs

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LITTLE MATHEMATICS LIBRARY
G.E.Shilov
PLOTTING
GRAPHS
Translated from the Russian
by
S. Sosinsky
MIR PUBLISHERS
MOSCOW

First published 1978
Ha aH2AUUCKOM flJblKe
.© English translation, Mir Publishers, 1978

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 where
graphs are not used. We have all often seen graphs, for example,
of industrial growth or increase in labour productivity. Natural
phenomena like daily or annual fluctuations of temperature or
atmospheric pressure are also easier to describe by means of graphs.
The plotting of graphs of that kind presents no difficulty, provided
the appropriate tables have been compiled. But we shall be
dealing here with graphs of a different kind, with graphs that
must be plotted from given mathematical formulas. A need for
such graphs often arises in various fields of knowledge. Thus, in
analysing the theoretical course of some physical process, a
scientist obtains a formula yielding some magnitude with which
he is concerned, for example, the amount of product obtained
relative to time. The graph plotted from this formula will provide
a clear picture of the future process. Looking at it, the scientist
may possibly introduce substantial changes into the scheme of
his experiment in order to obtain better results.
In this booklet we shall consider some simple methods of
plotting graphs from given formulas.
Let us draw two mutually perpendicular lines on a plane, one
horizontal and the other vertical, denoting their intersection by O.
II
We shall call the horizontal line the axis of abscissas and the
vertical the axis of ordinates. Each axis will be divided by the
point 0 into two semi-axes, positive and negative, the right half
of the axis of abscissas and the upper half of the axis of
ordinates being taken as positive, while the left half of the axis
of abscissas and the lower half of the axis of ordinates being
taken as negative. Let us mark the positive semi-axes by arrows.
The position of any point M on the plane can now be determined
by a pair of numbers. To find them we drop perpendiculars
from M to each of the axes; these perpendiculars intercept
segments 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 and
with a minus sign - when it is on the negative semi-axis, we call
5

the abscissa of point M and denote by x. Similarly, the length
of segment OB (with the same sign rule) we call the ordinate
of point M and denote by y. The two numbers x and yare
called the coordinates of point M. Every point on the plane has
certain coordinates. The points on the axis of abscissas have zero
ordinates and the points on the axis of ordinates have zero abscissas.
The origin of the coordinates 0 (the point of intersection of the
A
x
y
o
B --------IM
I
I
I
I
Fig. 1
axes) has both coordinates equal to zero. Conversely, when we are
given any two numbers x and y (negative or positive), we can
always plot a point M with abscissa x and ordinate y; to do that
we measure off a segment OA = x on the axis of abscissas and
draw a perpendicular AM = y at point A (bearing in mind the
sign); M will be the point we want.
Suppose we are given a formula for which we are to plot
the graph. The formula must indicate what operations must be
carried out on the independent variable (denoted by x) in order
to obtain the value we need (denoted by y). For example, the
formula
y = 1 + x2
indicates that the value of y can be obtained by squaring the
independent variable x, adding it to unity and dividing unity by
the result. If x assumes some numerical value .xo, then in
accordance with our formula y will assume some numerical value Yo.
The numbers Xo and Yo will determine a certain point M 0
in the plane of our picture. Instead of Xo we can take "some
other 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 the
plane. The locus of all points whose ordinates are related to their
abscissas by the given formula is called the graph of this formula.
The set of points on a graph, generally speaking, is infinite,
6

and we cannot hope in fact to plot them all without exception
according to the given rule. But we can manage without it.
In most cases it is enough to know a small number of points
in order to be able to judge the general form of the graph.
.,
o o
o
o
o
Fig. 2
The point-by-point plotting of a graph consists simply in
plotting a certain number of points of the graph and then joining
them (as far as possible) by a smooth curve.
As an example, let us consider the graph of the function
y = 1 + x2
Let us compile the following table:
(1)
x 0 1 2 3 -1 -2 -3
y
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 because
they are easier to operate with. In the second line we have
y
o
-------1~-~I___-_____f--~--_+--__+.--__+~ X
Fig. 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

(2)
y = (3x2 - 1)2
The point-by-point plotting, as we see, is extremely simple
and requires no 'theory' But, perhaps just because of that, we can
make bad mistakes by- blindly following it.
Let us plot the curve given by the equation
1
using this rule.
y
o o
Fig. 4
The table of values for x and y corresponding to this equation
is as follows:
x o 2 3 -1 -2 -3
y 1/6 76
The corresponding points on the plane are shown in Fig. 4.
The outline looks very like the previous one; joining the points
by a smooth line we obtain a graph (Fig. 5). Now it seems
o
Fig. 5
we can put down our pens and take a deep breath; we have
mastered 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 disagrees
with our picture. And there is no guarantee that in calculating
8

y for other intermediate values of x - and there is an in'inite
number of such values - we would not obtain even more striking
incongruities. Apparently the point-by-point plotting of a graph
itself is not well-founded.
*
* *
Now we shall consider another method of plotting graphs, more
reliable since it protects us from those unexpected things we have
just come across. This method - let us call it 'plotting by opera-
tions' - consists in carrying out directly on the graph all the
operations which are written down in the given formula: addition,
subtraction, multiplication, division, etc.
Let us consider some of the simplest examples. We plot the
graph 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= kx
with some coefficient k is obtained from the previous one by
multiplying each ordinate by the same number k. Suppose, for
example, k == 2; each ordinate of the previous graph must be
doubled. and .. as a result, we obtain a straight line more steeply
Fig. 7
the graph have equal abscissas and ordinates. The locus of points
having an ordinate equal to an abscissa is the bisector of the
angle formed by the positive semi-axes and the angle formed
by the negative semi-axes (Fig. 6). The graph corresponding to the
equation
2-19 9

rising upwards (Fig. 7). Each step to the right along the x..axis
corresponds to two steps along the y-axis. Incidentally, it is very
convenient to plot such graphs on squared or graph paper. In the
general case we will also get a straight line in the equation
y = kx. If k > 0, the line, with each step to the right, will move
up k steps along the y-axis. If k < 0, the line will slope down
instead.
----#-~---~x
Fig. 8
Now 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 b
units if b > 0; if b < 0, the original line will, of course,move
down instead. We will thus obtain a straight line parallel to the
initial one, but no longer passing through the origin of coordinates
and intercepting the segment b on the axis of ordinates (Fig. 8).
Therefore, the graph of any polynomial of degree one in x
is a straight line which can be plotted according to the above
rules.
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 =Yi
where
Yt = x
In other words, we will obtain the sought graph by squaring
10

each ordinate of the known line y = x. Let us see what we get in that
way.
Since 02 = 0, 12 = 1, (- 1)2 = 1, we have obtained three basic
points A, Band C (Fig. 9). When x > 1, Xl > x; therefore, to
the right of point B, the graph will lie above the bisector of
the quadrant angle (Fig. 10). When 0 < x < 1, 0 < x2
< x; therefore,
between points A and B the graph will lie below the bisector.
y
Co 08
A x
-1 0
Fig. 9
Moreover, we claim that when we approach point A, the graph
will be contained in any angle bounded from above by the straight
line y = kx (with an arbitrarily small k), and from below by the
x-axis; indeed, the inequality
Xl < kx
is satisfied only if x < k. This fact means that our curve is
tangent to the axis of abscissas at point 0 (Fig. 11). Now let
B/
/
/
/
A/ X
X
0 0
Fig. 10 Fig. 11
2* 11
y

us move along the x-axis to the left of point O. We know that
the numbers - a and + a give the same square (+ a2
). Thus
the ordinate of our curve for x = - a will be the same as for
x = +a. Geometrically this means that the graph of the curve
in the left half-plane will be a mirror image of the already
plotted 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. 13
Now by the same method we can plot a more complicated
curve
(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 rising
upwards (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 h
if h > 0 (Fig. 16). But if h < 0, we will have to move the curve
down instead (Fig. 17). All these curves are also called parabolas.
Now let us consider a somewhat more complicated example
of plotting a graph by multiplication. Suppose we are to plot
the graph according to the equation
y == x(x - I)(x - 2)(x - 3) (8)
12

y
Fig. 14
y
Fig. 15
Fig. 16
13

Here we are given the product of four factors. Let us plot
the graph of each of them separately: they are all straight lines,
parallel to the bisector of the quadrant angle and intercepting
the 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 zero
ordinate, since a product is equal to zero if at least one of
the factors is zero. At other points the product will be non-zero
Fig. 17
and will have a sign which can easily be found from the signs
of the factors. Thus, to the right of point 3 all the factors are
positive, therefore, so is their product. Between points 2 and 3 one
factor is negative, therefore, so is the product. Between points 1
and 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. 18
14

x, therefore, the product will also increase and very rapidly. To
the left of point 0 all the factors increase in the negative
direction, 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 multipli-
cation. Now let us consider division. Let us plot the curve
1
Y = t + x2
y
+ + +
x
0 2 3
(9)
Fig. 19
To do that we separately plot the graphs of the numerator and
the denominator. The graph of the numerator
Yt = 1
is a straight line parallel to the x-axis and passing through unity.
The graph of the denominator
Y2 == x
2
+ 1
y
Fig. 20
15

is a standard parabola, moved upwards by unity. Both of these
graphs are shown in Fig. 21.
Now we will carry out the division of each ordinate of the
numerator 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 the
denominator, so that the quotient is less than unity. Since the
numerator and the denominator are positive everywhere, the
quotient is positive, and, therefore, the graph is contained in the
strip bounded by the x-axis and the line y == 1. When x
infinitely increases, the denominator also grows infinitely, whereas
the numerator remains constant; therefore, the quotient tends to
zero. All this gives the following graph of the quotient (Fig. 22).
We have obtained the same picture as the one plotted previously
by the point-by-point plotting of a graph (p. 7}.
When carrying out division on the graph, particular attention
should be paid to those values for x at which the denominator
becomes zero. If the numerator is non-zero at this point, the
quotient becomes infinite. For example, let us plot the curve
1
y == ~ (10)
Here, the graphs of the numerator and the denominator are already
known (Fig. 23). For x = 1 we have Yl == Y2 == 1, which yields
Y == 1. When x > 1, the numerator is less than the denominator
and the quotient is less than unity as in the previous example.
When x increases infinitely, the quotient tends to zero and we
obtain the part of the graph which corresponds to the values
x > 1 (Fig. 24).
Let us consider now the values for x between °and 1. When x
moves from 1 to 0, the denominator tends to zero, while the
numerator remains equal to unity. Therefore, the quotient increases
infinitely and we obtain the branch moving up to infinity
(Fig. 25). When x < 0, the denominator as well as the whole
fraction become negative. The general form of the graph is presented
in Fig. 26. Now we can already start to plot the graph
discussed on page 8:
1
Y == (3x2 _ 1)2 (11)
Let us first plot the graph of the denominator. The curve
Y1 == 3x2
is a 'triple' standard parabola (Fig. 27). Subtracting unity
we move the graph down by unity (Fig. 28). The curve intersects
the x-axis at two points which will be easily found by setting
16

Numerator
o
Fig. 21
o
Fig. 22
y
Numerator
-------#---I------~x
Fig. 23
17

3x2
- 1 equal to zero:
Xl.2 = ± M= ±0.577...
Now let us square the obtained graph. At points Xl and X2
the ordinates remain equal to zero. All the other ordinates will
be positive, so that the curve will lie above the x-axis. At the
point x = 0 the ordinate will be equal to (- 1)2 = 1 and this
y
----I----+-----=l~x
Numerator
Fig. 24
y
Fig. 25
will be the greatest ordinate of the part of the graph between
Xl and X2. Outside this part the curve will steeply rise upwards
on both sides (Fig. 29).
Numerator
---+-----Jlr---
o
Numerator
~----+--I-----+----D~+---+----if-------:~X
..
Fig. 26
1.8

y
y
o
Fig. 27
----+---~--+---~x
Fig. 28
y
y
+--.JUL--+---..JIII~+-~X
-1
I
I
I
I
I
I
I Numerator
~~-+---
I
Denominator
o
Fig. 29 Fig. 30
19

The graph of the denominator has been plotted. In the same
drawing 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 quotient
will be positive, and the whole graph will lie above the x-axis.
When x = 0, the numerator and the denominator are equal, and
their quotient is equal to unity. Let us move along the x-axis
y
~
o
oc
·eo
e
II
Q
Numerator
Fig. 31
to the right from point O. The numerator remains equal to unity
and the denominator decreases; therefore, the quotient increases
beginning with unity. When we reach the value X2 = 0.577...
the denominator will become equal to zero. This means that the
quotient will become infinite by this moment (Fig. 30). Beyond
the point X2 the denominator will quickly move in the opposite
direction from the zero value to unity and will then infinitely
increase. On the contrary, the quotient will return from infinity
to unity intersecting the straight line y" = 1 at the same point as Y3
and further will indefinitely approach zero (Fig. 31).
The same picture will be obtained to the left of the axis of
ordinates (Fig. 32).
We have marked on this graph the points corresponding to
the integral values for x = 0, 1, 2, 3, -1, -2, - 3. These are
the same points which have been taken in the point-by-point
plotting of a graph on page 8. But the actual form of the graph
20

y
Fig. 32
Fig. 33
----1--IS---+-+--+----......,f---+---+-+--+----It---+---+-+--+--t--~x
Fig. 34
21

radically differs from the one proposed in Fig. 5. We see that
actually instead of smoothly decreasing from the value of 1 (for
x = 0) to the value of 1/4 (for x = 1) and further, the curve
moves upwards to infinity. Here we can see the point with
coordinates x = 1/2, y = 16 which had no place on the previous
incorrect graph but which gets in quite nicely on the new correct one.
*
* *Summarizing the general rules which should be applied to plot
graphs 'by operations', we can say:
(a) All the operations contained in the given formula must
be carried out with the graphs going from simple ones to more
complicated.
(b) When multiplying graphs pay attention to the points where
factors (at least one of them) become zero; remember the rule of
signs between those points.
(c) When dividing graphs pay attention to the points where
denominator vanishes. If at those points numerator is non-zero,
the branches of the curve will move to infinity - up or down
depending on the signs of the numerator and the denominator.
(d) Pay attention to the behaviour of the curve when x
moves indefinitely to the right (to +(0) or to the left (to - oo].
Here we have only described the simplest operations which
may be carried out with graphs. To be more precise, we started
out with the simplest equation y = x and applied further the four
operations of arithmetic: addition, subtraction, multiplication, and
division. To these operations we could easily add the algebraic
operation - extracting roots.
But more complicated operations - trigonometric and logarithmic
ones - can also be carried out with graphs. One only needs to
know the graphs of the simplest equations y = sin x (Fig. 33)
and y = logx (Fig. 34). Using the methods desribed above we can
plot 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 be
able to answer many natural questions arising when one considers
some graphs. For example, we see on a certain graph that the
curve rises to some value Yo, then begins to move down; we say
in 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 our
limited scope of methods. Other questions arise, for example, at
what angle the curve intersects the x- or the y-axis, whether it
22

is convex upward or downward. Our methods cannot give exact
answers to all of those questions. Here, a considerably more
serious knowledge of mathematical technique is required. The
methods which allow the study of the above properties of graphs
are contained in the branch of mathematics known as the differential
calculus.
*
* *
In conclusion solve some problems on graphs.
Plot the graphs of the following equations:
1. y = x2
+ X + 1. 3. y = x2
(x - 1).
2. y = x(x2
- 1).
x
5. Y=--l.
x-
4. y = x(x - 1)2
x 1
Hint. Single out the integral part: ---= 1 + --.x-I x-I
x2
6. y= --1.
x-
Hint. Single out the integral part.
x 3
7. y = --1-.
'x-
Hint. Single out the integral part.
8. y = ± ~.
Hint. The square root of negative numbers does not exist In
the real part.
9.y=±~
How to prove that this curve is a circle?
Hint. Recall the definition of a circle and the Pythagorean
theorem.
1ft y=±~.
23

Prove that the upper branch of the curve indefinitely approaches
the bisector of the quadrant angle when x ~ 00.
Hint. Vx2
+ 1 - x == 1
~+x·
11. y= ±xVx(l-x).
12. y= ±X2~.
13. y= 1~.
2 ± V1 - x2
14. Y == x 2/3 (1 - x)2/3 .

Answers to.Problems
y
2
To Problem
~-+---+---+---~x
-2 -1
To Problem 2
y
To Problem 3 To Problem 4
25

y I
I
t
I
I
I
I
I
I
-~----------I
I
26
To Problem 5
To Problem 6

y
1)1I I
I I
I J
I I
I I
: I.....
I 1"1-
I ' :
I Itf
t '/tI I :a.-
,I
)'
/ I
I I
/ I
I I
I
I
il
I
I
I
I
I
y
To Problem 7
To Problem 8

--+----+---~-~x
To Problem 9
------~-----~x
To Problem to
y
o- -....~::-------+---......x
To Problem 11
28

To Problem 12
y
1
o
.....IK-------+-----..a....-~x
-1
To Problem 13
o
To Problem 14
29

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

TO THE READER
Mir Publishers welcome your comments on the content,
translation and design of this book.
We would also be pleased to receive any proposals you
care to make about our future publications.
Our address is:
USSR., 129820., Moscow, 1-110, GSP
Pervy Rizhsky Pereulok, 2
Mir Publishers
Printed in the Union of Soviet Socialist Republics

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