Applying graphs
By Clement Radcliffe, Contributor

AS PROMISED last week, I will begin to review graphs and their applications in our lesson today. I do expect that you are comfortable with straight line graphs.

GRAPHS

Prior to plotting a straight line or a curve, the following should be noted.

* The y and x axes must be clearly labelled.

* The scale given must be used exactly, or if no scale is given, an appropriate one should be used.

* The use of a suitable pencil (HB) is required.

Straight line graphs should be drawn with a ruler, while a curve must be drawn free hand.

* No matter the shape of the curve, under no circumstance should a ruler be used.

* Be very careful in completing the table of values if they are not all given as, unfortunately, you will be penalised for any error you make.

* The existence of deviation by any point from the straight line or the smooth curve is an indication that an error has been made. You must immediately review your calculation.

Let us examine the following example.

Given the equations:

(a) y = 2x - 3 (-3 < x < 3)
(b) y = x² - 3x + 2 (3 < x < 3)

1. Prepare the table of values with respect to the domain given.
2. Plot both graphs

APPLICATIONS

* Solution of Simultaneous Linear Equations

If two linear equations are plotted, their point of intersection represents the solution of the simultaneous equations.

Given the equation 2x - y = -1 and 3x - y = 2, the point x = 3 and y = 7 is common to both and indeed is the point of intersection. It is therefore the solution of the simultaneous equations.

Please verify this by plotting the lines. You may also verify the answers by substitution.

* Solution of Simultaneous Linear Equations, One Linear and One Quadratic

The principle is similar to the above, but first you need to be able to plot quadratic graphs.

PLOT OF QUADRATIC GRAPHS

Plot the graph y = 2x² + x -3 for real values of x in the domain: -3 < x <2. In doing so, please ensure that:

(a) The tables of values are completed accurately.

(b) The points are clearly shown and are joined by free hand. (No ruler, please).

Having plotted the graph, it may then be used to solve the simultaneous equations.

EXAMPLE

Using the same axes, plot the equations:

y = 2x² + x - 3; y = 1 - x, and hence solve the two equations simultaneously.

Completing the tables:

y = 2x (2) + x -3

y = 1 -x

The points of intersection are: (-2, 3) and (1,0)

* The solutions are: x = 2, y = 3 and x = 1, y = 0.
* Solution of Quadratic Equations

You may use the quadratic graph plotted above to solve quadratic equations. As the graph y = 2x² + x - 1 was plotted, then you may solve the equation 2x² + x - 1 = 0.

In this case, 2x² + x - 1 = 0 is equivalent to y = 2x² + x - 1 and y = 0.

The points of intersection of the curve (y = 2x² + x - 1) and the line (y = 0) or the x axis represents the solution.

Please refer to the graph to find the solution for the equations above.

From the graph also the solution of the equation 2x² + x - 3 = 1 - x is therefore x = -2, y = 3 and x = 1, y = 0.

By simplifying the equation, the values may be shown also to be the solution of 2x² + 2x - 4 = 0 as well as x² + x -2 = 0.

HOMEWORK

Solve graphically the simultaneous equations:

y = 3x(2) - 2x - 1 and y = x + 5.

Clement Radcliffee is Principal of Glenmuir High School in Clarendon. Send your questions and comments to the CXC Study Guide, the Gleaner Company Ltd., 7 North Street, Kingston; or email us at jcampbell@gleanerjm.com.