CHAPTER XII

GRAPHICAL REPRESENTATION OF THE SOLUTIONS OF SYSTEMS OF SIMULTANEOUS QUADRATIC EQUATIONS

455. Graph of the General Quadratic Function ax2 + bx + c.

This problem is illustrated by the following example:

EXAMPLE. Plot the equation y = x2 - 4 x — 5.

In the table below are arranged the various values of y which correspond respectively to values x = 0, +1, +2, :

— 1, 2, 3, etc., in the equation y = x2

Y

+10

4x

NOTE 1.-It is clear that in case the values of x increase beyond those given in the table, each corresponding value of y will be larger than the one preceding; hence the values of x and y given in the table are sufficient to determine the ultimate directions of the curve. In general, when this is found to be the case, one need not compute more values for the table.

NOTE 2.-If the graph of an equation of the second degree in two variables consists of a single branch which extends to a part of the plane at an infinite distance from the origin, it is called a parabola.

456. TYPE I.

(2441.)

Graphs of curves of the quadratic form

Ax2 + 2 Вxy + Cy2 + 2 Dx + 2 Ey + F = 0, of the straight line ax + by + c = 0 (8245), and of the location of the points determined by their solutions.

EXAMPLE.-Plot the curves represented by the equations

and the points (243) represented by the solutions of these equations.

Plotting the points in the first table gives the line BAPST, Fig. 2. Plotting the points in the second table gives the curve R'Q'P'APQR. Plotting the points in the third table gives the eurve on the left, LA'M, which is in every respect equal to the curve on the right. The curves LA'M and R'AR are called the branches of the graph represented by the equation y = — 16. *

When the graph of a curve of the second degree in two variables consists of two branches, each of which extends to infinity, the graph is called an hyperbola. For example, y2x2-16 is the equation of an hyperbola (Fig. 2).

In reckoning the values of y which correspond to the values of x, in case of the hyperbola y2 = x2 16, we notice that for one value of x there correspond two values of y which are equal and opposite in sign. The same is true for the values of x which correspond to a value of y. For this reason the hyperbola is said to be symmetrical to the x- and y-axis.

which are represented by the points A and P respectively.

NOTE. The points corresponding to the imaginary results x = 0, y = ±√16, etc., are not situated on the hyperbola.

457. TYPE II. (8442.)

Graphs of curves of the quadratic forms ax2 + bxy+cy2 = d and Ax2 + Bxy + Cy2 = D, and of the location of the points determined by their solutions.

Plotting the points corresponding to the plus values of x and both plus and minus values of y in the second table, we get the branch curve BPAQB'; similarly, for the minus values of x and the corre

sponding plus and minus values of y we get the branch curve BSA'RB'. These two branches make up the entire curve BAB'A'B which is the graph of the first equation.

Similarly, by plotting the values of x and y in the first table, we get the graph of the second equation, the hyperbola whose branches are PA,Q and SAR. The points corresponding to the imaginary values of x and y are not points on either of these curves. The solutions of this system of simultaneous quadratic equations are

Jx1=2√3=3.46.. {x1⁄2= 3.46.. § x3=−3.46 . . § x4=−3.46 . . | y1=21/2=2.82 . . y2=-2.82..

Y3 2.82..

14=-2.82.. The points corresponding to these pairs of values of x and y are the intersections of the two graphs (Fig. 3), namely, P, Q, R, S.

=

NOTE 1.-In case the graph of an equation of the second degree in two variables is a closed curve, the graph is called an ellipse. For example, 12x2 + 13y2 248 is the equation of an ellipse.

NOTE 2-In case the equation has the form x2 + y2 : = 16 42, the curve is a circle whose radius is 4.

and

458. Graphs of miscellaneous quadratic forms.

EXAMPLE. -Plot the equations,

The first equation may be written

(1) y3 2 (x + 4) y + (x2.

4x

xy=

2.

SOLVING EQUATION (1)

y = x+4±2v 3 (x+3)

*The student should note that, in equation (2), when x is-, y is +; and that, as x approaches 0 through negative values, y is + and approaches +. Similarly, when isy is; and as a approaches 0 through positive values, y approaches. Thus as a passes through 0 from positive to negative values, y changes sign from - to +∞.