The Slope, Everywhere at Once

You'll learnto build a slope field out of one substitution repeated, to find its spine before plotting anything, and to settle a matching question with a coordinate pair instead of an impression.

A slope field answers one question at every point of the plane: if a solution passed through here, how steep would it be? The equation answers it with arithmetic, one point at a time, and the picture that results shows the whole family of solutions before a single one has been found. Two habits do most of the work. Find where the slope is zero first — that set is the spine every other segment leans away from. And when two candidate equations produce fields that look alike, stop looking and substitute a point where they disagree.

8 STEPS · 6 QUICK CHECKS · SUBSTITUTE THE POINT, DO NOT JUDGE THE PICTURE · v1

dy/dx = x + y ⇒ zero slope on y = −x ⇒ solutions y = Ce ˣ − x − 1

Before you start

What you're looking at

A grid of points in the plane, each carrying a short segment. The segment's slope is whatever dy/dx works out to at that point, so the whole picture is one substitution repeated.

The question

Given an equation, where do the segments lie flat, which way do they lean on either side of that, and does the answer change when you move sideways, upward, or both?

Watch for

The horizontal and vertical scales are equal in every panel here, so a slope of 2 really does look like a slope of 2. On a stretched window it would not.

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