Slope Fields — Visualizing Solutions

A slope field shows tiny line segments at grid points, each with slope = dy/dx at that point.

How to read them:

Follow the segments like a river current. Solutions curve along the flow.

How to draw them:

Given dy/dx = x + y:

At (0,0): slope = 0 (horizontal segment)

At (1,0): slope = 1 (45-degree segment)

At (0,1): slope = 1

At (1,-1): slope = 0

Plot segments at each point.

Matching slope fields to equations:

If dy/dx depends only on x → all segments in each column are parallel

If dy/dx depends only on y → all segments in each row are parallel

If dy/dx = 0 along a line → segments are horizontal there

AP exam: Multiple choice will show slope fields and ask you to match them to differential equations.

Reference:

Wikipedia: Slope Field

https://en.wikipedia.org/wiki/Slope_field