__2.2 Solving equations graphically__The solution of the equation

*f*(*x*) =*g*(*x*) can be solved by graphical method.**: Draw the graphs of**

*Step 1**y*=

*f*(

*x*) and

*y*=

*g*(

*x*) on the same axes.

**: The points of intersection of the graphs are the solutions of the equation**

*Step 2**f*(

*x*) =

*g*(

*x*). Read the values of

*x*from the graph.

**Solution of an Equation by Graphical Method**

**Example 1:**

**(a)**The following table shows the corresponding values of

*x*and

*y*for the equation

*y*= 2

*x*

^{2}–

*x*– 3.

x |
–2 |
–1 |
–0.5 |
1 |
2 |
3 |
4 |
4.5 |
5 |

y |
7 |
m |
– 2 |
–2 |
3 |
12 |
n |
33 |
42 |

Calculate the value of

*m*and*n*.**(b)**

*For this part of the question, use graph paper. You may use a flexible curve rule.*

By using a scale of 2cm to 1 unit on the

*x*-axis and 2cm to 5 units on the*y*-axis, draw the graph of*y*= 2*x*^{2}–*x*– 3 for –2 ≤*x*≤ 5.**From your graph, find**

(c)

(c)

**(**The value of

*i*)*y*when

*x*= 3.9,

**(**The value of

*ii*)*x*when

*y*= 31.

**(d)**Draw a suitable straight line on your graph to find the values of

*x*which satisfy the equation 2

*x*

^{2}–

*3*

*x*= 10 for –2 ≤

*x*≤ 5.

Solution:Solution:

**(a)**

*y*= 2

*x*

^{2}–

*x*– 3

when

*x*= –1,**= 2 (–1)**

*m*^{2}– (–1) – 3 =

**0**

when

*x*= 4,**= 2 (4)**

*n*^{2}– (4) – 3 =

**25**

(b)

(b)

**(c)**

**(**From the graph, when

*i*)*x*= 3.9,

*y*=

**23.5**

**(**From the graph, when

*ii*)*y*= 31,

*x*=

**4.4**

**(d)**

*y*= 2

*x*

^{2}–

*x*– 3 —– (1)

2

*x*^{2}–*3**x*= 10 —– (2)*y*= 2

*x*

^{2}–

*x*– 3 —– (1)

0 = 2

*x*^{2}–*3**x*– 10 —— (2) ←**(Rearrange (2))**(1) – (2) :

*y*= 2*x*+*7*The suitable straight line is

*y*= 2*x*+*7.*Determine the

*x*-coordinates of the two points of intersection of the curve*y*= 2

*x*

^{2}–

*x*– 3 and the straight line

*y*= 2

*x*+

*7.*

x |
0 |
4 |

y = 2x + 7 |
7 |
15 |

From the graph,

*x*= –1.6, 3.1