But a word on gradient.
Take a graph.
Any graph.
Put a straight line in it.
Any angle.
Take the left hand side of the line. What are its co-ordinates? In the picture above, the line begins at point (1,1). From that point, call it A, the line goes along and up. As you move along the line its co-ordinates rise in value. It moves along the x-axis and the y-axis.
Gradient is simply a measure of how y changes as you change x values. In other words, steepness.
Look at point B. It is at (5,4). From A to B you have gone 4 x-units along and 3 y-units up. On this line, wherever you are on the graph, if you go 4 x units along you will ALWAYS go 3 y units up.
Gradient, which is a general statement of this quantity, is calculated through this difference between the co-ordinates at A and B.
We divide the difference in y across the two points, by the difference in x.
y2-y1/x2-x1, meaning: the y co-ordinate of point B minus the y co-ordinate of point A, then divided by the same thing done for the x co-ordinates.
In this case,
= 4-1/5-1
= 3/4
The gradient is "three quarters".
In the picture below the line is slanting downwards. This will give it a negative gradient. Same method though.
Horizontal lines have a gradient of 0.
Vertical lines are a tricky one. If you think of a line going straight up and down, and try to apply this logic to it, you end up with a weird division.
Say you've got a vertical line through points (5,0) and up to (5, 5). If you do the y2-y1/x2-x1 calculation to find the gradient, you end up doing 5/0. You can't divide by 0, so you can't get a result. There isn't a gradient for a vertical line.
Another way of looking at it is that a vertical line cannot be a result of y = f(x) because it would be one-many - ie it's not a function.
2 comments:
Heh. In my defence, I did pretty much know what it meant - I Just had to look it up to make sure. But a great post nonetheless.
If you're aiming to write a blog accessible even to people with my poor grasp of mathematics then may I suggest the use of sockpuppets. :-)
Thanks, Matt.
I don't know how to draw sockpuppets: even graphs is proving tricky! But if I did know, I probably would.
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