Gradient is another word for slope. The slope (or gradient) of a line is the vertical change divided by the horizontal change. We also say it is "rise over run". Here are some examples describing how the gradient is worked out. Blue line (f(x)) The line can be seen to go through the (x, y) points (0, -1) and (1, 1). The vertical difference between these two points (rise) is (1 - (-1)) = 2. The horizontal difference (run) between these two points is (1 - 0) = 1. Rise/run = 2/1 = 2. The gradient of the blue line is 2. (You can see this also because 2 is the coefficient of x in the equation for f(x).) Red line (g(x)) The line can be seen to go through the points (0, 2) and (3, 1). The rise between these two points is (1-2) = -1. The corresponding run is (3-0) = 3. The slope (gradient) is rise/run = -1/3, which we can see is the coefficient of x in the equation for g(x). 
