34 Chapter 2Algebraic functions
A frame of referenceis defined in the plane, consisting of a fixed point called the
originof the coordinate system, and two perpendicular axes (directed lines), called
the coordinate axes, which intersect at the origin. In Figure 2.2 the origin is labelled
O, the coordinate axes are the xand yaxes, and the plane is called the xy-plane.
The position of a point in the plane is specified by the ordered pair (x, y), where xis
the x-coordinateor abscissaand yis the y-coordinateor ordinate. A point with
coordinates (x, y) lies at perpendicular distance|x|from the y-axis and|y|from the
x-axis. It lies to the right of the (vertical) y-axis ifx 1 > 10 and to the left ifx 1 < 10 ; it lies
above the x-axis ify 1 > 10 and below ify 1 < 10. The origin has coordinates (0, 0).
In an actual example, suitable scales are marked on the coordinate axes, and each
pair of numbers in a table such as Table 2.1 is plotted as a point on the graph. If
the function is known to vary smoothly between the plotted points (as it is in this
example) then the points may be joined by a smooth curve. The curve is the graphical
representation of the function.
2.3 Factorization and simplification of expressions
The structure of an algebraic expression can often be simplified and clarified by the
process of factorization (see Section 1.3 for factorization of numbers). For example, in
the expression
3 xy 1 + 16 x
2each term of the sum can be written as the product of the common factor ( 3 x) and
another term:
3 xy 1 + 16 x
21 = 1 (3x) 1 × 1 (y) 1 + 1 (3x) 1 × 1 (2x)
Therefore
3 xy 1 + 16 x
21 = 1 (3x) 1 × 1 (y 1 + 12 x) 1 = 13 x(y 1 + 12 x)
and the algebraic expression has been written as the product of the two factors ( 3 x)
and (y 1 + 12 x).
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origi nO
x-axis
y-axis
y
x
y
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(x,y)
xy-plane
Figure 2.2