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Revision:Gradients

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Gradients

1 Finding the gradient of a straight-line graph
- 1.1 Example
2 Finding the gradient of a curve
- 2.1 Example
3 Comments

Finding the gradient of a straight-line graph

It is often useful or necessary to find out what the gradient of a graph is - that is, a measure of how steep the line is.

For a straight-line graph, you need to pick two points on the graph which you know the coordinates for.

The gradient of the line = (change in y-coordinate)/(change in x-coordinate)

So if you pick 2 points P_1 = (x_1, y_1) and P_2 = (x_2, y_2) then the gradient is:

$\displaystyle \frac{y_1 - y_2}{x_1 - x_2}$ or $\frac{\Delta y}{\Delta x}$ .

Note: It does not matter which set of coordinates come first, provided it is the same for both x and y.

Example

Find the gradient of the line joining the points (1, 5) and (4, 11).

$\displaystyle \frac{y_1 - y_2}{x_1 - x_2} = \frac{5-11}{1-4} = \frac{-6}{-3} = 2$

We can then use this to find the equation of the line using the formula:

y - y_1 = m(x - x_1)

where m is the gradient.

In this case it gives (either coordinate can be used):

$y - 5 = 2(x-1) \Rightarrow y - 5 = 2x-2$

which can be rearranged to get:

y = 2x+3

Finding the gradient of a curve

It is also possible to find the gradient of a curve at a given point - thought the gradient (steepness) will change as you move along the curve, so your answer will only be true for the gradient at the point you use.

To find the gradient of a curve, you must draw an accurate sketch of the curve. At the point where you need to know the gradient, draw a tangent to the curve. A tangent is a straight line which touches the curve at one point only. You then find the gradient of this tangent like you did in the section above.

Example

Find the gradient of the curve y = x^2 at the point (3, 9).

(Image missing illustrating the graph and how to find the gradient at the given point graphically.)

Note: this method only gives an approximate answer. The better your graph is, the closer your answer will be to the correct answer. If your graph is perfect, you should get an answer of 6 for the above question.