Basic Engineering Mathematics, Fifth Edition

Graphical solution of equations 157

(a)

y

y 5 x 2

x

2

1

21 0 1

(b)

y 53 x 2

y

x

2

1

21 0 1

y

x

2

1

21 0 1

(c)

y 5 21 x 2

Figure 19.4

Graphs ofy=−x^2 ,y=− 3 x^2 andy=−

1

2

x^2 are

shown in Figure 19.5. All have maximum values

at the origin (0, 0).

(a)

y

y 52 x 2

21 x

21

22

0

1

(b)

y 523 x 2

21

22

y

21 x

0

1

21

22

y

(^21) x
0
1
(c)
y 52 21 x 2
Figure 19.5
Wheny=ax^2 ,
(i) curves are symmetrical about they-axis,
(ii) the magnitude ofaaffects the gradient of the
curve, and
(iii) the sign ofadetermines whether it has a
maximum or minimum value.
(b) y=ax^2 +c
Graphs of y=x^2 + 3 ,y=x^2 − 2 ,y=−x^2 + 2
andy=− 2 x^2 −1 are shown in Figure 19.6.
Wheny=ax^2 +c,
(i) curves are symmetrical about they-axis,
(ii) the magnitude ofaaffects the gradient of the
curve, and
(iii) the constantcis they-axis intercept.
(c) y=ax^2 +bx+c
Wheneverbhas a value other than zero the curve
is displaced to the right or left of they-axis.
Whenb/ais positive, the curve is displacedb/ 2 a
to theleft of they-axis, as shown in Figure19.7(a).
Whenb/a is negative, the curve is displaced
b/ 2 a to the right of they-axis, as shown in
Figure 19.7(b).
Quadratic equationsof the formax^2 +bx+c= 0
may be solved graphically by
(a) plotting the graphy=ax^2 +bx+c,and
(b) noting the points of intersection on thex-axis (i.e.
wherey=0).
y
y 5 x 213
y 52 x 212
y 522 x 221
y 5 x 222
210 x
(a)
3
1
y
x
22
21 1
(b)
2
0
y
21 x
(c)
0
2
1
y
21 x
21
24
(d)
0
1
Figure 19.6
(a)
25242322211
y
x
12
10
6
4
2
0
y 5 x 216 x 111
(b)
y 5 x 225 x 14
21
y
0 x
2
4
6
22
1234
8
Figure 19.7
The x values of the points of intersection give the
required solutions since at these points bothy=0and
ax^2 +bx+c=0.
The number of solutions, or roots, of a quadratic equa-
tion depends on how many times the curve cuts the
x-axis. There can be no real roots, as in Figure 19.7(a),
one root, as in Figures 19.4 and 19.5, or two roots, as in
Figure 19.7(b).
Here are some worked problems to demonstrate the
graphical solution of quadratic equations.
Problem 3. Solve the quadratic equation
4 x^2 + 4 x− 15 =0 graphically, given that the
solutions lie in the rangex=−3tox=2.
Determine also the co-ordinates and nature of the
turning point of the curve