The equation to find the gradient of a straight line is:
Gradient=y1 – y2
X1 – x2
The gradient of a straight line is a constant, the gradient of a curve is not. To find the gradient of a curve we need to choose certain points along the curve, e.g. at x=1, x=2…etc. We then draw a tangent from these points and work out the gradient of the tangent.
I am going to start by looking at the equation ‘y=axn’ .
When:
a=1 n=1 then the equation is Y=x
a=2 n=1 then the equation is Y=2x
In the equation Y=ax¹ the gradient is ‘a’
Y=X2
I am going to look at the line y=x2 first:
(See graph, fig.1)
From the graph we can see that as the values of the coordinates increases, so does the gradient. We can see this from the results below and the graph.
There are two methods for finding the gradient of a line, as can be seen from the table above, I have already explained the ‘tangent’ method.
The small increment method is done by using calculations rather than values from the graph, two points are chosen, the two ‘x’ value’s are chosen at random, we then find the ‘y’ value’s by squaring the ‘x’ value. The closer the two points the more accurate the result will be, for example if we choose ‘2’ as the first ‘x’ value we could then choose ‘2.001’ as the next ‘x’ value. The gradient is found using the equation:
Gradient=y1-y2
x1-x2
So using the example numbers from above I would find the gradient as shown below:
Gradient=y1-y2
x1-x2
=4.004001-4
2.001-2
Gradient =4.001
This method is more accurate than using the results from the graph. This is because when taking results from the graph it is very difficult to get them as precise.
