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Sketching
Slope Fields

Learn to visualize differential equations by drawing tiny line segments that reveal the direction of solutions at every point in the plane.

AP Calculus AB

What Is a Slope Field?

A slope field (also called a direction field) is a graphical representation of a first-order differential equation of the form \(\dfrac{dy}{dx} = f(x, y)\). At each sample point \((x, y)\) in the plane, you draw a short line segment whose slope equals \(f(x, y)\).

Visualizing \(\dfrac{dy}{dx} = \)\(x\)

Why Do Slope Fields Matter?

Unsolvable DEs
Many differential equations have no closed-form solution. Slope fields let us see the behavior anyway.
Qualitative Analysis
Identify equilibrium solutions, growth/decay patterns, and long-term behavior at a glance.
AP Exam Staple
The AP Calculus AB exam regularly asks you to sketch slope fields and match them to equations.

The Core Rule

The collection of all these segments forms the slope field. Any solution \(y(x)\) to the DE must be tangent to the field at every point it passes through.

Slope Depends on \(x\), \(y\), or Both

\(f(x)\) only
Segments in the same column are parallel. Example: \(\dfrac{dy}{dx} = x\)
\(f(y)\) only
Segments in the same row are parallel. Example: \(\dfrac{dy}{dx} = y\)
\(f(x, y)\)
Slopes vary in both directions. Example: \(\dfrac{dy}{dx} = x + y\)

Slope Field vs Solution Curve

Slope Field Solution Curve
What it shows Direction at every sample point One particular path through the field
Initial condition Not needed Required: a specific \((x_0, y_0)\)
Uniqueness One field per DE Infinitely many: one per initial condition
How to draw Compute slopes, draw segments Follow the segments from the starting point

Using Isoclines to Sketch Faster

An isocline is a curve along which the slope \(f(x, y)\) is constant. By finding isoclines, you can draw many parallel segments at once.

1
Set \(f(x, y) = c\) for a chosen constant \(c\). Solve for the isocline curve.
2
Draw the isocline on the plane. Every point on this curve has slope \(c\).
3
Draw segments with slope \(c\) along the isocline. Repeat for several values of \(c\).

How to Sketch a Slope Field

1
Choose a grid of sample points: typically integer coordinates in a region like \([-3, 3] \times [-3, 3]\).
2
Evaluate \(f(x, y)\) at each point to get the slope \(m\).
3
Draw a short segment centered at each point with the computed slope. Keep all segments the same length.
4
Look for patterns: horizontal segments (\(m = 0\)), steep regions, symmetry, and isoclines.
5
Sketch solution curves by following the flow of the segments from chosen initial points.

Example 1: \(\dfrac{dy}{dx} = \)\(y\)

Example 2: \(\dfrac{dy}{dx} = \)\(-\dfrac{x}{y}\)

Example 3: \(\dfrac{dy}{dx} = \)\(x + y\)

Spotting Equilibrium Solutions

An equilibrium solution is a constant solution \(y = k\) where \(\frac{dy}{dx} = 0\) for all \(x\). On a slope field, it appears as a horizontal line of flat segments.

Stable
Nearby solution curves approach the equilibrium. Segments point toward it from both sides.
Unstable
Nearby curves diverge away. Segments point away from the equilibrium line.

Common Mistakes to Avoid

Wrong Variable
Substituting \(x\) where \(y\) should go (or vice versa) when evaluating \(f(x, y)\). Always double-check which coordinate you are using.
Unequal Lengths
Drawing longer segments for steeper slopes. All segments should have the same length: only the angle should change.
Crossing Segments
Making segments so long they overlap or cross, creating visual confusion. Keep segments short and centered at each grid point.

Summary Cheat Sheet

Observation What It Means
Horizontal segments \(f(x, y) = 0\) at those points
Vertical segments \(f(x, y)\) is undefined (or \(\pm\infty\))
Same slope in each column DE depends only on \(x\)
Same slope in each row DE depends only on \(y\)
Segments form closed loops Solutions may be circles or ellipses
Row of flat segments Equilibrium solution \(y = k\)

Question 1 of 5

If \(\dfrac{dy}{dx} = 3x\), what is the slope of the segment at the point \((2, 5)\)?

Question 2 of 5

In a slope field for \(\dfrac{dy}{dx} = y\), all segments in the same ____ have identical slopes.

Question 3 of 5

Which equation produces a slope field where every segment along the line \(y = x\) is horizontal?

Question 4 of 5

For the DE \(\dfrac{dy}{dx} = y(1 - y)\), which values of \(y\) give equilibrium solutions?

Question 5 of 5

An isocline for the DE \(\dfrac{dy}{dx} = 2x + y\) with slope constant \(c = 4\) is the curve: