Consider z as abscissa, y as ordinate, and construct the cubic
parabola y = z\

Through (3, 0) and (-1, 0)
draw ordinates and let them meet
the curve in A and G. On the
ordinates lay off downward AB
= 12, and CD = 12, and draw BD.
By measuring the abscissas of

the points of intersection, we

obtain the roots :

2=2, 1, and — 3.
Hence a; = — 1, —2, and — 6.
T






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54



ORAPHIC ALGEBRA



On OX lay off 00' = 3.
62. In the preceding diagram z represents tlie abscissas, bnt
by changing the location of the y-a,xis we can obtain abscissas
which equal x.
Consider 0' as the new origin,
and the ordinate ToTJ, drawn
tlirough 0', as the new y-asis.
Then the abscissa of any point
is smaller by 3 than the old ab-
scissa z, or the new abscissa is
a — 3, i.e. x. By thus introduc-
ing a new axis, the entire work
of the preceding paragraph can
be done without introducing z at all.
Thus, instead of saying :

If «-3= -4,(^-3)^ + 9 (2-3)^ + 20(2-3)-+- 12 = 12,
we have briefly :

If a;=-4, ar' + 9a;^ + 20a; + 12 = 12.
Similarly, instead of measuring the z, we may directly meas-
ure the X, and thus obtaiu the roots of (1).
63. A change in the position of the axes is called a trans-
formation of coordinates.
