I have found the gradients for all of the numbers shown, I did this by drawing a triangle as close to the graph line as I could. I then find the height and the width of the triangle. Then I would divide the height by the width to find the gradient.
The following is an example….
If the height of the triangle 3 = 12
And the width of triangle 3 = 3
Therefore…….12/3 = 4
The gradient of point (3,9) is 3.
This table shows my results
The difference between each one is obviously 2, so my formula must be.. Gradient = 2x
I will now show the gradients of chords starting at the point (2,4) and finishing at various other points along the plotted line for example, 2 to 5 and 2 to 4 etc.
The pattern is seen.
There is a formula is the A level textbook ‘Essential Pure Mathematics’
The formula is
RQ = NQ – NR
= (a + h)² - a²
= 2ah + h²
and also
PR = h
So the gradient of the chord PQ is
RQ = 2ah + h²
PR h
= h(2a + h)
h
= 2a + h
As h gets smaller it would be stupid to add it onto the formula, so this proves that my original formula was correct as 2a is now left in the formula.
