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.

:)