Lesson

Reading a slope field as a set of orders

Each tiny segment is the slope the DE demands at that point.

Learning goals

Match a slope field to a differential equation.
Sketch a solution curve through a given starting point.

Explanation

A slope field is a grid of little marks. At (x, y), the mark’s slope is the value of dy/dx given by the DE. If dy/dx depends only on y, the marks in a horizontal row match. If it depends only on x, a vertical column matches. That visual test is often faster than solving.

A solution curve must follow the marks, like a boat following current arrows. Through a given initial point there is typically one curve you are meant to draw. Crossing the marks at the wrong angle means you stopped obeying the DE.

Key terms

Slope field. A plot of short segments whose slopes equal dy/dx from a differential equation at sample points.
Particular solution. A solution that also satisfies a stated initial condition.

Common mistakes

Drawing a solution curve that ignores the nearby segment slopes.
Confusing the field for dy/dx = y with the field for dy/dx = x.

Practice

Original Marlow Works items. Check the answer explanation after you try.

1 of 2Limits and Continuity · easy · multiple choice

What is lim (x→3) of (x² − 9) / (x − 3)?

Original Marlow Works item — not a College Board question.

Take your time—this is practice, not a test.

Answer explanation

In the field for dy/dx = y, why are the marks flat along the x-axis?

Answer. When y = 0, the DE says the slope is 0, so those segments are horizontal.

The field is literally plotting the right-hand side.

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Related resources

External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.

Reference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source