A slope field is a diagram showing the gradient prescribed by a differential equation at many points in the coordinate plane. You interpret it by following the direction of the small line segments to predict the shape and behaviour of particular solution curves.
This is IB Mathematics: Applications and Interpretation HL-only content from AHL 5.15: Slope fields.
How a slope field works
For a first-order differential equation
the value of gives the gradient at the point ((x,y)). A short line segment with that gradient is drawn there. Repeating this across a grid produces the slope field.
For example, if
then at , the gradient is
The segment at therefore slopes upward with gradient . At , the gradient is , so the segment is horizontal.
| Segment pattern | Interpretation |
|---|---|
| Positive gradient | Solutions increase as increases. |
| Negative gradient | Solutions decrease as increases. |
| Zero gradient | The solution has a horizontal tangent; a line of zero gradients may indicate an equilibrium solution. |
| Large gradient magnitude | Solutions change steeply. |
A solution curve through an initial point must pass through that point and remain locally tangent to the field's segments. The field therefore shows a family of possible solutions, while an initial condition selects one particular solution.
A common misconception is that the individual segments form the solution. They do not: they only indicate local direction. Your curve should be smooth and follow those directions rather than join the segments like separate line pieces.
Exam technique
In an HL exam, you may be asked to sketch a solution through a given point or interpret long-term behaviour. Ensure the curve passes through the stated point, follows the segment directions, and identifies features such as increasing regions, decreasing regions, or equilibrium behaviour. This applies to GDC-active HL paper contexts, especially extended-response and investigation questions.