7.3 Sketching Slope Fields

What a slope field shows

A slope field (direction field) is a grid of short line segments, each showing the slope of a solution curve to a differential equation dy/dx = f(x,y) at that point. At each (x, y), you plug into f(x,y) to get the slope, then draw a tiny segment with that slope. Solution curves follow the flow of these segments.

Sketching a slope field

Pick several grid points (x, y). At each point, compute the slope using dy/dx = f(x,y), then draw a short segment with that slope centered at the point. Segments with slope 0 are horizontal; steep positive/negative slopes lean steeply up/down. Look for patterns like symmetry or where slopes equal zero (horizontal tangent lines).

Matching curves to fields

A particular solution curve must be tangent to the slope segments it passes through at every point — it never crosses a segment at an angle. To match a solution to a slope field, trace a smooth curve that stays parallel to nearby segments, checking that it passes through any given initial condition point.

Key terms

Slope field

A grid of short line segments showing the slope dy/dx = f(x,y) of solution curves at many points.

How to draw a slope field segment

Compute f(x,y) at a grid point, then draw a short segment with that slope centered there.

Horizontal segments in a slope field occur where...

f(x,y) = 0, meaning dy/dx = 0 at those points.

Relationship between solution curves and slope field

A solution curve must be tangent to the slope field segments at every point it passes through.

Using an initial condition with a slope field

Start at the given point and sketch a curve that stays tangent to nearby segments to trace the particular solution.

Try an AP-style question

For dy/dx = x + y, what is the slope segment at (1, 2)?

1. −1
2. 2
3. 1
4. 3