Saturday, 16 August 2008

Straight Line Graph Conversion

The mathematical expression for a straight line is y = mx + c,
where m is the gradient and c is the intercept on the vertical axis (y).

When a straight line is plotted, we get a simple straight line cutting across the y axis at the point indicated by the constant "c" in the general straight line equation.

Diagram 1: A straight line graph y = 2x + 1 with x-axis as "x".

What if the equation is y = 2x2 + 1 ?
Is it still a straight line when the horizontal axis is "x"? Let us see the diagram below.

Diagram 2: y = 2x2 + 1 is not a straight line when the horizontal axis is "x".

Now, let us replace the "x2" in the expression with "z" and see what happens to the graph when plotted with the horizontal axis as "z".

Diagram 3: A straight line graph with horizontal axis plotted as "z".

From diagram 3, we can see that by replacing the original "x2" with "z", we can convert the parabolic curve of diagram 2 to a straight line graph.

This is so because the equation y = 2 x2 + 1 is now expressed as y = 2 z + 1 , a straight line general format!

In other words, when we plot y = 2x2 + 1 in the graph format "y against x2", we should expect a straight line graph.

Let's do another example:
y = 2/x + 3

Plotting this against horizontal axis of "x" gives the below graph.

Diagram 4: Graph when plotted with horizontal axis as "x".

Now, when we substitute the horizontal axis with (1/x), we should be getting a straight line graph following the principle we applied in the previous example.

Diagram 5: Straight line graph with horizontal axis plotted as "1/x".

When we are able to convert any mathematical expression to the straight line format
y = mx + c, and apply the horizontal scale according, we can reduce the complexity of graphical curve to a simple straight line, which will be easy enough for analysis.

When concepts and principle of mathematical operations are understood, a complex problem can be reduced to a simple question as with the graphical technique of above. After all, a straight line is still much easier to handle than a curve with varying gradient.

:)