7.4 Reasoning Using 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 at that point, computed from the differential equation dy/dx = f(x,y). Plugging any point's coordinates into f(x,y) gives the slope of the segment drawn there. Sketching a solution means drawing a curve that follows the segments' directions at every point it passes through.

Sketching solution curves

To sketch a particular solution, start at the given initial condition point and trace a smooth curve that stays tangent to the nearby slope segments as you move left and right. The curve should never cross segments at an odd angle — it must flow with the local slopes throughout its path, matching the field's pattern (e.g., flat spots where slope = 0, steep spots where slope is undefined or large).

Matching equations to fields

Given a slope field, you can identify horizontal tangent lines (where dy/dx = 0) and vertical tangent lines (where slope is undefined) by checking where f(x,y) equals zero or is undefined. This helps match a differential equation to its correct field, or verify that a proposed solution curve is consistent with the field's segments.

Key terms

Slope field

A grid of short segments showing the slope dy/dx = f(x,y) at each point, used to visualize solution curves.

How to find a segment's slope

Substitute the point's (x, y) coordinates into f(x,y) from the differential equation.

Sketching a solution curve

Start at the initial condition and draw a curve that stays tangent to nearby slope field segments.

Where segments are horizontal

Where dy/dx = f(x,y) = 0.

Where segments are vertical (undefined)

Where f(x,y) is undefined, often where a denominator equals zero.

Try an AP-style question

Which differential equation matches?

1. dy/dx = x − 2
2. dy/dx = 2 − y
3. dy/dx = 2y
4. dy/dx = y − 2