Unit 7.3 – Sketching Slope Fields

AP® Calculus AB & BC | Visualizing Solutions to Differential Equations

Why This Matters: Slope fields (also called direction fields) provide a powerful visual way to understand differential equations without solving them algebraically! They show the "flow" of solutions and help you sketch solution curves. This graphical approach is essential for understanding the behavior of solutions and appears frequently on AP® exams. Master slope fields and you'll gain intuition about differential equations that goes beyond formulas!

🎯 What is a Slope Field?

DEFINITION

A slope field is a visual representation of a differential equation \(\frac{dy}{dx} = f(x, y)\) that shows small line segments at various points \((x, y)\) in the plane. Each segment has slope equal to \(f(x, y)\).

Key Concept:

At point \((x, y)\), draw a small segment with slope \(\frac{dy}{dx} = f(x, y)\)

📝 Think of it this way: If a solution curve passes through point \((x, y)\), the slope field shows what direction that curve is heading at that point!

🔍 Why Use Slope Fields?

Slope Fields Allow You To:

Visualize solutions: See the general behavior without solving
Sketch solution curves: Draw approximate solutions through any point
Understand long-term behavior: See where solutions go as \(x \to \infty\)
Identify equilibrium solutions: Horizontal lines where slope = 0
Match DEs to graphs: Common AP® exam question type

✏️ How to Sketch a Slope Field

Step-by-Step Process

Creating a Slope Field by Hand:

Start with the differential equation: \(\frac{dy}{dx} = f(x, y)\)
Create a grid of points: Choose several \((x, y)\) values
Calculate slopes: For each point, compute \(f(x, y)\)
Draw segments: At each point, draw a short line with that slope
- Positive slope: segment tilts up /
- Zero slope: horizontal segment —
- Negative slope: segment tilts down \
- Undefined/large slope: nearly vertical |
Keep segments short: They should all be roughly the same length
Look for patterns: Points with same slope form curves

📖 Detailed Example: Creating a Slope Field

Example 1: \(\frac{dy}{dx} = x\)

Problem: Sketch the slope field for \(\frac{dy}{dx} = x\)

Solution:

Step 1: Analyze the equation

Notice: slope depends only on \(x\), NOT on \(y\)!

This means all points with the same \(x\)-coordinate have the same slope.

Step 2: Calculate slopes at sample points

\(x\)	\(y\)	Slope = \(x\)	Direction
-2	any	-2	Steep down \\
-1	any	-1	Down \
0	any	0	Horizontal —
1	any	1	Up /
2	any	2	Steep up /

Step 3: Key observations

At \(x = 0\): all segments are horizontal
For \(x > 0\): segments slope upward
For \(x < 0\): segments slope downward
All segments in a vertical line have the same slope

Visual Description:

Imagine vertical columns of segments:

Left side (\(x < 0\)): segments tilt down like \\\\\\

Center (\(x = 0\)): segments horizontal ———

Right side (\(x > 0\)): segments tilt up like ///

Solution curves: These would be parabolas \(y = \frac{x^2}{2} + C\)

Example 2: \(\frac{dy}{dx} = y\)

Problem: Describe the slope field for \(\frac{dy}{dx} = y\)

Analysis:

Slope depends only on \(y\), NOT on \(x\)!

All points with the same \(y\)-coordinate have the same slope.

Key features:

At \(y = 0\): all segments horizontal (slope = 0)
For \(y > 0\): segments slope upward (positive slope)
For \(y < 0\): segments slope downward (negative slope)
As \(|y|\) increases, slopes become steeper
All segments in a horizontal line have the same slope

Pattern:

Horizontal rows of segments, steepness increases away from \(y = 0\)

Solution curves: exponential functions \(y = Ce^x\)

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

Problem: Sketch the slope field for \(\frac{dy}{dx} = x + y\)

Analysis:

Slope depends on BOTH \(x\) and \(y\)!

Sample calculations:

Point \((x, y)\)	Slope = \(x + y\)	Direction
(0, 0)	0	Horizontal —
(1, 0)	1	Up /
(0, 1)	1	Up /
(1, 1)	2	Steep up /
(-1, 0)	-1	Down \
(0, -1)	-1	Down \

Key observation:

Points where \(x + y = 0\) (line \(y = -x\)) have horizontal segments

Points where \(x + y = k\) (constant) have same slope

👁️ Reading and Interpreting Slope Fields

✅ What to Look For:

Isoclines: Curves along which slope is constant
- Find by setting \(\frac{dy}{dx} = k\) (constant)
Zero-slope isocline: Where segments are horizontal
- Set \(\frac{dy}{dx} = 0\) to find
Equilibrium solutions: Horizontal solution curves (where \(\frac{dy}{dx} = 0\) for all \(x\))
Direction of flow: How solutions move as \(x\) increases
Behavior at infinity: Where solutions go as \(x \to \infty\)

✏️ Sketching Solution Curves on Slope Fields

Drawing Solution Curves

How to sketch a solution through a point:

Start at the given point
Follow the slope field: Draw curve tangent to segments
Move smoothly: Curve should "flow" with the field
Don't cross segments: Curve must be tangent, not cut through
Continue in both directions: Unless reaching a boundary

💡 Key Principle: A solution curve passing through \((x_0, y_0)\) must be tangent to every segment it touches in the slope field!

🔬 Special Slope Field Patterns

Common Patterns to Recognize:

Differential Equation Types and Their Slope Fields
DE Form	Pattern	Example
\(\frac{dy}{dx} = f(x)\)	Vertical columns (same slope in each column)	\(\frac{dy}{dx} = x\)
\(\frac{dy}{dx} = g(y)\)	Horizontal rows (same slope in each row)	\(\frac{dy}{dx} = y\)
\(\frac{dy}{dx} = k\) (constant)	All segments have same slope	\(\frac{dy}{dx} = 2\)
\(\frac{dy}{dx} = ky\)	Exponential growth/decay pattern	\(\frac{dy}{dx} = 0.5y\)
\(\frac{dy}{dx} = k(y - a)\)	Horizontal line at \(y = a\)	\(\frac{dy}{dx} = y - 2\)

🎯 Matching DEs to Slope Fields

🔥 Strategy for Matching:

Check for horizontal segments: Find where \(\frac{dy}{dx} = 0\)
Check vertical/steep segments: Where is slope undefined or very large?
Test specific points: Calculate slope at easy points like (0,0), (1,0), (0,1)
Look for symmetry: Is field symmetric about axes?
Identify pattern: Does slope depend on x, y, or both?
Check equilibrium: Horizontal solution curves?

Quick Test Points Strategy:

For matching problems, test these points quickly:

(0, 0): Origin—easiest calculation
(1, 0): On x-axis
(0, 1): On y-axis
(1, 1): First quadrant

If slopes match at 2-3 points, likely correct match!

❌ Common Mistakes to Avoid

Mistake 1: Making segments too long (they should be short and uniform)
Mistake 2: Not keeping segments centered on the point
Mistake 3: Solution curves that don't follow the field
Mistake 4: Solution curves that cross each other (impossible!)
Mistake 5: Forgetting that horizontal segments mean slope = 0
Mistake 6: Confusing steep positive vs steep negative slopes
Mistake 7: Not checking sign of slope (positive/negative)
Mistake 8: Assuming all DEs produce nice symmetric fields
Mistake 9: Drawing curves through wrong points
Mistake 10: Not using test points to verify matching

📝 Practice Problems

For each DE, describe key features of its slope field:

\(\frac{dy}{dx} = 2x\)
\(\frac{dy}{dx} = -y\)
\(\frac{dy}{dx} = x - y\)
\(\frac{dy}{dx} = 1\)

Answers:

Vertical columns; horizontal at \(x=0\); slopes increase with \(|x|\)
Horizontal rows; horizontal at \(y=0\); negative slopes for \(y>0\), positive for \(y<0\)
Horizontal segments along line \(y=x\); depends on both variables
All segments have slope 1 (parallel lines at 45°)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Sketching slope fields: Segments centered on points, appropriate slopes
Drawing solution curves: Curves tangent to slope field
Matching problems: Clear reasoning (test specific points)
Identifying features: Know where slope = 0, equilibrium solutions
Using slope fields: To predict behavior of solutions

💯 Exam Strategies:

For sketching: Draw 9-15 segments neatly, focus on accuracy
For matching: Test 2-3 key points to eliminate wrong answers
For solution curves: Make sure they're smooth and tangent to segments
Time management: Don't spend too long making it perfect
Use a ruler/straightedge: If allowed, makes segments neater

⚡ Quick Reference Guide

SLOPE FIELD ESSENTIALS

The 5-Step Sketching Process:

✓ Choose grid points
✓ Calculate slope at each point: \(\frac{dy}{dx} = f(x,y)\)
✓ Draw short segment with that slope
✓ Keep segments uniform length
✓ Look for patterns

Key Indicators:

Horizontal segments: Slope = 0
Vertical columns: \(\frac{dy}{dx}\) depends only on \(x\)
Horizontal rows: \(\frac{dy}{dx}\) depends only on \(y\)
Solution curves: Must be tangent to every segment

Master Slope Fields! A slope field visualizes a differential equation \(\frac{dy}{dx} = f(x,y)\) by drawing small line segments with appropriate slopes at grid points. To sketch: (1) choose points, (2) calculate slope at each point, (3) draw short segment with that slope, (4) keep segments uniform. Key patterns: if slope depends only on \(x\), vertical columns of identical segments; if only on \(y\), horizontal rows. Isoclines are curves where slope is constant. To sketch solution curves, follow the slope field smoothly—curves must be tangent to segments. For matching problems, test specific points like (0,0), (1,0), (0,1). Find equilibrium solutions by setting \(\frac{dy}{dx} = 0\). Common on AP® exams: sketching parts of slope fields, matching DEs to fields, drawing solution curves. Solution curves never cross (uniqueness theorem). Practice visualizing—slope fields give intuition about DE behavior without solving! 🎯✨