Slope fields
Also known as: direction field
A slope field is a graph of short line segments drawn at grid points, each showing the slope a differential equation assigns to that point. It lets you visualize the family of solution curves without solving the equation.
A slope field (or direction field) is a visual representation of a first-order differential equation. At each point (x, y) on a grid, you draw a short line segment whose slope equals the value of dy/dx at that point. The result is a field of dashes that shows, at a glance, how solution curves flow through the plane.
Sketching one is mechanical: evaluate the differential equation at each grid point and draw a segment with that slope. For dy/dx = x + y, the segment at (1, 1) has slope 2, the segment at (0, 0) is horizontal, and the segment at (1, −1) is horizontal too, since 1 + (−1) = 0. A solution curve through a given initial condition is traced by following the segments — each segment is tangent to the solution passing through it.
Slope fields matter because many differential equations are hard or impossible to solve analytically, yet their slope fields still reveal the behavior of solutions: where they increase or decrease, where they level off, and whether they approach an equilibrium.
On the AP Calculus AB exam, slope field questions rarely ask you to solve anything. Instead they ask you to sketch segments at selected points, match a slope field to its differential equation, or match it to a solution curve. Fast checks help: find where the slope is zero, determine whether slopes depend only on x (segments identical in vertical columns) or only on y (identical in horizontal rows), and test the sign of the slope in each quadrant.
Key takeaways
- A slope field draws a short segment at each grid point with slope equal to dy/dx there.
- Solution curves follow the segments — each segment is tangent to the solution through that point.
- If dy/dx depends only on x, segments match down each column; if only on y, they match across each row.
- To match a slope field to its equation, check where slopes are zero and the sign of the slope in each region.
- AP Calculus AB tests slope fields through sketching, matching, and initial-condition reasoning rather than solving.