42 Chapter 2Algebraic functions
and
(2.13)
defines the constant slope. The line crosses the horizontal x-axis at one point:
y 1 = 1 0 when (2.14)
This value of xis called the rootof the linear function. In general, the roots of a
polynomial function are those values of the variable for which the value of the function
is zero; that is, the roots are the solutions of the polynomial equation
f(x) 1 = 10 (2.15)
EXAMPLE 2.14Find the equation of the straight line that passes through the points
(− 1 , − 6 ) and (3, 2).
Let the line bey 1 = 1 mx 1 + 1 c. Then:
at point(x
1,y
1) 1 = 1 (−1, −6), − 61 = 1 −m 1 + 1 c
at point(x
2, y
2) 1 = 1 (3, 2), 21 = 13 m 1 + 1 c
Solution of the pair of simultaneous equations (see Section 2.8) givesm 1 = 12 and
c 1 = 1 − 4. Therefore
y 1 = 12 x 1 − 14
The graph of the line is shown in Figure 2.7. The line has slopem 1 = 12 , which means
that the value of yincreases twice as fast as that ofx. The line crosses the y-axis at
y 1 = 1 c 1 = 1 − 4 , and crosses the x-axis atx 1 = 12.
0 Exercises 40–43
x
c
m
=−
m
yy
xx
=
−
−
2121.............................................................................................................− 1 0 12 3
x
y
− 2
− 4
2
4
(3,2)
(− 1 ,−6)
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Figure 2.7