Gradient

In mathematics, a gradient is the rate a function increases.

If a function has value <math>f(a)</math> at <math>x=a</math> and <math>f(b)</math> at <math>x=b</math> with <math>a<b</math> than the gradient of <math>[a,b]</math> is,

<math>\frac{F(b)-F(a)}{b-a}</math>.

In schooling, usually early high school, students are taught that the gradient is equal to "the rise over run" or more formally,

<math>\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}</math>,

the rise being defined as the difference between the highest point and the lowest point, negative if the highest is on the left and positive if the highest point in on the right. The run is defined as difference of the right <math>x</math>-value and the left <math>x</math>-value (see Cartesian coordinates).

This concept is usually first introduced with the introduction of linear equations. The equation of a straight line in Cartesian coordinates is given by,

<math>y=mx+c</math>

where <math>m</math> is the gradient of the line and <math>c</math> is the value of the <math>y</math> coordinate when <math>x=0</math>, this is called the y-intercept, e.g, where the line intercepts the y-axis.