01Cgradient

= Gradient =

The ** gradient ** of a graph is the measure of how steep the graph is.

The word gradient is used with the same meaning in English when
 * we describe how steep a railway track is by discussing the gradient of the track
 * we describe how steep a rock face is for rock climbers

We always read a graph from left to right .. (the same direction as we read English)
 * If the y-values of a graph are going up as we move from left to right, the gradient is positive


 * If the y-values of a graph are not changing as we move from left to right, the gradient is zero


 * If the y-values of a graph are going down as we move from left to right, the gradient is negative.


 * Rise over Run **

The gradient of a line between two points can be calculated by using:

... where
 * ** Rise ** = the change in the y-values between the two point
 * how far the graph has gone __**up**__ between the two points[[image:bhs-methods12/01Criserun.gif align="right"]]
 * if the graph has gone down, write it as a negative
 * ** Run ** = the change in the x-values between the two points
 * how far the graph has gone __**across**__ between the two points.


 * Example 1 **

Find the gradient of the graph shown to the right


 * Solution:**

math \\ .\qquad \text{gradient } = \dfrac{\text{rise} }{\text{run}} \\. \\ . \qquad \qquad \qquad = \dfrac{ \text{distance up} }{\text{distance across}} \\. \\ . \qquad \qquad \qquad = \dfrac{2}{5} math


 * Gradient Rule **

The same rule can be expressed more mathematically like this.

The gradient (m) between two points is given by:
 * (x 1, y 1 ) .. and
 * (x 2, y 2 )




 * Example 2 **

Find the gradient of a line joining the points (3, 4) and (5, – 6)


 * Solution:[[image:01Ceg2.gif align="right"]]**

... First Point math . \qquad \Big( x_1 = 3, \;\; y_1 = 4 \Big) math

... Second Point math . \qquad \Big( x_2 = 5, \;\; y_2 = -6 \Big) math

... ** Gradient ** math \\ . \qquad m = \dfrac{y_2 - y_1}{x_2 - x_1} \\. \\ . \qquad \quad = \dfrac{-6 - 4}{5 - 3} \\. \\ . \qquad \quad = \dfrac{-10}{2} \\. \\ . \qquad \quad = -5 math


 * Note: **
 * Because we always read graphs from left to right
 * The first point should always be the point on the left
 * The second point should be the point on the right
 * This particular rule will still give the correct gradient if you accidentally swap the two


 * Gradient and Trigonometry **

If define the angle (q ) between the line and the __positive__ x-axis.

Then we get this diagram:

Recall that: math . \qquad \tan \big( \theta \big) = \dfrac{\text{opposite}}{\text{adjacent}} math

Then from the diagram, we get:

math . \qquad \tan \big( \theta \big) = \dfrac{ \text{rise}} { \text{run}} math

Hence the gradient (m) is also given by:


 * Example 3 **

Find the angle a line with a gradient of 2.5 forms with the positive x-axis.


 * Solution:**

math \\ . \qquad \tan \big( \theta \big) = 2.5 \\. \\ . \qquad \theta = \tan^{-1} \big( 2.5 \big) \\. \\ . \qquad \theta = 68.2^\circ math


 * Example 4 **

Find the gradient of the line shown here (round to 2 decimal places):

... ... To use our rule for gradient, ... ... we need the value of q , ... ... which is the angle to the __positive__ x-axis ... ... (see diagram to right)
 * Solution:**

... ... Using geometry, we know that

math \\ . \qquad \theta = 180 - 65 \\. \\ . \qquad \theta = 115^\circ math

hence math \\ . \qquad m = \tan \big( \theta \big) \\. \\ . \qquad m = \tan \big( 115 \big) \\. \\ . \qquad m = -2.14 math

Notice that tan(q ) automatically makes the gradient negative for a line sloping down.

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