Straight line graphs 135

(a)

8 7 6 5 4 3 2 1

2110

AB

C

y

234

(b)

11

10

8

6

4

2

24232221

F

ED

x 0 x

y 523 x (^12) y
y 52 x 11
y
2
1
0 23
3
(c)
1 x
y 53
Figure 17.8
In Figure 17.8(a), a straight line graphy= 2 x+1is
shown. To find the gradient of this straight line, choose
two points on the straight line graph, such asAandC.
Then construct a right-angled triangle, such asABC,
whereBCis vertical andABis horizontal.
Then, gradient ofAC=
change iny
change inx

CB
BA

7 − 3
3 − 1

4
2
= 2
In Figure 17.8(b), a straight line graphy=− 3 x+2is
shown. To find the gradient of this straight line, choose
two points on the straight line graph, such asDandF.
Then construct a right-angled triangle, such asDEF,
whereEFis vertical andDEis horizontal.
Then, gradient ofDF=
change iny
change inx

FE
ED

11 − 2
− 3 − 0

9
− 3
=− 3
Figure 17.8(c) shows a straight line graph y=3.
Since the straight line is horizontal the gradient
is zero.
17.4.2 They-axis intercept
The value ofywhenx=0 is called they-axis inter-
cept. In Figure 17.8(a) they-axis intercept is 1 and in
Figure 17.8(b) they-axis intercept is 2.
17.4.3 The equation of a straight line graph
The general equation of a straight line graph is
y=mx+c
wheremis the gradient andcis they-axis intercept.
Thus, as we have found in Figure 17.8(a),
y= 2 x+1 represents a straight line of gradient 2 and
y-axis intercept 1. So, given the equationy= 2 x+1,
we are able to state, on sight, that the gradient= 2
and they-axis intercept=1, without the need for any
analysis.
Similarly, in Figure 17.8(b),y=− 3 x+2representsa
straight line of gradient−3andy-axis intercept 2.
In Figure 17.8(c),y=3 may be rewritten asy= 0 x+ 3
and therefore represents a straight line of gradient 0 and
y-axis intercept 3.
Here are some worked problems to help understanding
of gradients, intercepts and equations of graphs.
Problem 2. Plot the following graphs on the same
axes in the rangex=−4tox=+4 and determine
the gradient of each.
(a)y=x (b)y=x+ 2
(c)y=x+5(d)y=x− 3
A table of co-ordinates is produced for each graph.
(a) y=x
x − 4 − 3 − 2 − 1 0 1 2 3 4
y − 4 − 3 − 2 − 1 0 1 2 3 4
(b) y=x+ 2
x − 4 − 3 − 2 − 1 0 1 2 3 4
y − 2 − 1 0 1 2 3 4 5 6
(c) y=x+ 5
x − 4 − 3 − 2 − 1 0 1 2 3 4
y 1 2 3 4 5 6 7 8 9
(d) y=x− 3
x − 4 − 3 − 2 − 1 0 1 2 3 4
y − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1
The co-ordinates are plotted and joined for each graph.
TheresultsareshowninFigure17.9.Eachofthestraight
lines produced is parallel to the others; i.e., the slope or
gradient is the same for each.