23. Polynomials

Further examples of functions and their graphical representation.

The examples which follow will give the reader a better notion of the infinite variety of possible types of functions.

A.   Polynomials. A polynomial in \(x\) is a function of the form \[a_{0}x^{m} + a_{1}x^{m-1} + \dots + a_{m},\]

where \(a_{0}\), \(a_{1}\), …, \(a_{m}\) are constants. The simplest polynomials are the simple powers \(y = x\), \(x^{2}\), \(x^{3}\), …, \(x^{m}, \dots\). The graph of the function \(x^{m}\) is of two distinct types, according as \(m\) is even or odd.

First let \(m = 2\). Then three points on the graph are \((0, 0)\), \((1, 1)\), \((-1, 1)\). Any number of additional points on the graph may be found by assigning other special values to \(x\): thus the values \[\begin{aligned} {6} x &= \tfrac{1}{2},\quad &&2,\quad &&3,\quad &&-\tfrac{1}{2},\quad &&-2,\quad &&3 \\ \text{give} \quad y &= \tfrac{1}{4},\quad &&4,\quad &&9,\quad &&\tfrac{1}{4},\quad &&4,\quad &&9.\end{aligned}\] If the reader will plot off a fair number of points on the graph, he will be led to conjecture that the form of the graph is something like that shown in fig. 8. If he draws a curve through the special points which he has proved to lie on the graph and then tests its accuracy by giving \(x\) new values, and calculating the corresponding values of \(y\), he will find that they lie as near to the curve as it is reasonable to expect, when the inevitable inaccuracies of drawing are considered. The curve is of course a parabola.

There is, however, one fundamental question which we cannot answer adequately at present. The reader has no doubt some notion as to what is meant by a continuous curve, a curve without breaks or jumps; such a curve, in fact, as is roughly represented in fig. 8. The question is whether the graph of the function \(y = x^{2}\) is in fact such a curve. This cannot be proved by merely constructing any number of isolated points on the curve, although the more such points we construct the more probable it will appear.

This question cannot be discussed properly until Ch. V. In that chapter we shall consider in detail what our common sense idea of continuity really means, and how we can prove that such graphs as the one now considered, and others which we shall consider later on in this chapter, are really continuous curves. For the present the reader may be content to draw his curves as common sense dictates.

It is easy to see that the curve \(y = x^{2}\) is everywhere convex to the axis of \(x\). Let \(P_{0}\), \(P_{1}\) (fig. 8) be the points \((x_{0}, x_{0}^{2})\), \((x_{1}, x_{1}^{2})\). Then the coordinates of a point on the chord \(P_{0}P_{1}\) are \(x = \lambda x_{0} + \mu x_{1}\), \(y = \lambda x_{0}^{2} + \mu x_{1}^{2}\), where \(\lambda\) and \(\mu\) are positive numbers whose sum is \(1\). And \[y – x^{2} = (\lambda + \mu)(\lambda x_{0}^{2} + \mu x_{1}^{2}) – (\lambda x_{0} + \mu x_{1} )^{2} = \lambda\mu(x_{1} – x_{0})^{2} \geq 0,\] so that the chord lies entirely above the curve.

The curve \(y = x^{4}\) is similar to \(y = x^{2}\) in general appearance, but flatter near \(O\), and steeper beyond the points \(A\), \(A’\) (fig. 9), and \(y = x^{m}\), where \(m\) is even and greater than \(4\), is still more so. As \(m\) gets larger and larger the flatness and steepness grow more and more pronounced, until the curve is practically indistinguishable from the thick line in the figure.

The reader should next consider the curves given by \(y = x^{m}\), when \(m\) is odd. The fundamental difference between the two cases is that whereas when \(m\) is even \((-x)^{m} = x^{m}\), so that the curve is symmetrical about \(OY\), when \(m\) is odd \((-x)^{m} = -x^{m}\), so that \(y\) is negative when \(x\) is negative. fig. 10 shows the curves \(y = x\), \(y = x^{3}\), and the form to which \(y = x^{m}\) approximates for larger odd values of \(m\).

It is now easy to see how (theoretically at any rate) the graph of any polynomial may be constructed. In the first place, from the graph of \(y = x^{m}\) we can at once derive that of \(Cx^{m}\), where \(C\) is a constant, by multiplying the ordinate of every point of the curve by \(C\). And if we know the graphs of \(f(x)\) and \(F(x)\), we can find that of \(f(x) + F(x)\) by taking the ordinate of every point to be the sum of the ordinates of the corresponding points on the two original curves.

The drawing of graphs of polynomials is however so much facilitated by the use of more advanced methods, which will be explained later on, that we shall not pursue the subject further here.