**Line graphs**:

If you are looking at a line graph, the gradient is 'rise over run'. If you count a number of squares up on the vertical axis (y axis) and then count the number the graph has moved along the horizontal axis (x axis, often 'time') in that time (forwards or backwards, depending if the slope is positive or negative), then the gradient is the number of squares up on the vertical axis divided by the number of squares along on the horizontal axis.

If you have the equation of the line graph, which in standard form is

y = ax + b

the number 'a' is the gradient of the slope of the line, and 'b' is the intercept of the line (where the line cuts the vertical or y axis). In other words, the gradient is the coefficient that accompanies the x variable.

On distance-time graphs, the gradient of the line equals the speed (or velocity if there is a direction of travel). if the distance is in km and the time in seconds, then the speed is measured in km/s (distance units divided by time units), which is read as km per second or kms^-1.

**Line graphs and beyond:**

For more complex graphs in general, the gradient is the rate of change of the graph in the y direction with respect to change in the x direction. The rate of change can be established by differentiating the function of the graph f(x) to get the derivative function f'(x). For linear graphs (of order 1), x to the power 1 differentiates to 1, since the rate of change is constant (every movement on the x axis corresponds to the same amount of change each time on the y axis). For higher order or non-linear functions, for example a parabola, the rate of change is no longer constant. In the case of a parabola (involving an x^2 term), the rate of change increases (or decreases) linearly in x, since the derivative of x^2 is 2x (multiply by the old power and decrease the old power by one to get the new power). Like a lot of maths, rules on differentiation need to be memorized as there are different rules for all the different types of non-linear functions. Some graphs have places where there is no slope or derivative as it takes no value for that particular value of x, or there is a discontinuity, where the graph might shoot of the page, or move in steps.