Topic 7.3 Notes – Sketching Slope Fields

What a slope field is

A slope field (also called a direction field) represents a differential equation of the form

$$\frac{dy}{dx} = f(x,y)$$

At each point $(x,y)$, the equation tells you the slope of the solution curve passing through that point.

So you:

• Pick a point $(x,y)$.
• Plug it into $f(x,y)$.
• Draw a short line segment with that slope centered at the point.

Do that for many points on a grid. The result is a “field” of little segments.

Every actual solution curve must:

• Be tangent to those segments at every point.
• Follow the general direction they indicate.

Here’s what that looks like for the differential equation $\frac{dy}{dx} = x - y$:

Slope field for $\frac{dy}{dx} = x - y$ with solution through (0,1)

The red curve is one specific solution passing through $(0,1)$. Notice how it hugs the tiny line segments everywhere it goes. That’s the whole idea.

How to sketch a slope field

On a quiz, you won’t usually calculate 30 slopes. You look for structure.

Suppose you’re given

$$\frac{dy}{dx} = 2x - y$$

Here’s the thinking process:

1. Identify the slope formula
   The slope at $(x,y)$ is $2x - y$.
2. Check for zero slopes first
   Set $2x - y = 0 \Rightarrow y = 2x$.
   Along that line, all segments are horizontal. That line often organizes the whole field.
3. Plug in a few easy points
   • At (0,0): slope = 0
   • At (1,0): slope = 2
   • At (0,1): slope = -1
4. Draw short, consistent-length segments
   Only the angle changes, not the length.

Patterns matter more than quantity. A neat, thoughtful sketch scores better than a messy dense one.

Recognizing common patterns

Before computing anything, ask: what does the equation depend on?

When $\frac{dy}{dx} = f(x)$

Slopes depend only on x.

• All points with the same $x$-value have the same slope.
• Slopes form vertical stripes.

Here’s a classic example with $\frac{dy}{dx} = x$. Notice how every vertical line has matching segments.

Slope field for $\frac{dy}{dx} = x$

If you see vertical bands of identical segments, the equation probably depends only on $x$.

When $\frac{dy}{dx} = f(y)$

Slopes depend only on y.

• All points with the same $y$-value share the same slope.
• Slopes form horizontal bands.

Now look at $\frac{dy}{dx} = y$. The segments repeat across horizontal lines.

Slope field for $\frac{dy}{dx} = y$

This shows up a lot with growth/decay models.

Zero and undefined slopes

• If $f(x,y) = 0$, you draw horizontal segments.
• If the slope would be undefined (like $\frac{x}{y}$ when $y=0$), you don’t draw a segment there. That line often acts like a boundary.

AP-style multiple choice questions love asking where solutions level off. Look for rows of nearly horizontal segments.

Estimating solution curves

If you’re given an initial condition like $y(-1)=2$:

• Start at the point $(-1,2)$.
• Follow the direction of nearby segments.
• Draw a smooth curve that stays tangent as it moves.

Important observations you should be able to make from the field:

• Increasing vs decreasing
  Positive slopes → curve rises.
  Negative slopes → curve falls.
• Leveling off
  Near zero slopes, solutions flatten.
• Solutions don’t cross
  Through any single point, there is only one possible slope. So solution curves never intersect each other.

On FRQs, when asked to sketch a particular solution, make sure your curve actually matches the tiny segments. Readers look for tangency.

Why slope fields matter

Some differential equations are hard or impossible to solve explicitly. A slope field still tells you:

• Long-term behavior
• Stability near certain lines
• Whether solutions increase, decrease, or approach a boundary

You’re learning to understand behavior without solving algebraically. That modeling mindset is the point of Unit 7.