Sketching Slope Fields: Study Guide (Calculus AB 7.3)

Course scope. This topic is shared by Calculus AB and Calculus BC. Everything here is examinable for both.

A note on notation

A differential equation gives the derivative of an unknown function. In this unit it usually has the form dy/dx = (an expression in x and y), for example dy/dx = x − 2y. You may also see y′ for dy/dx, or other letters such as dH/dt when the variables have a meaning. Coordinates are written (x, y), so the first number is always the x-value.

What a slope field shows

Take dy/dx = x − 2y. You do not know the solution function y yet. But the equation still tells you something useful at every point. Suppose a solution curve passes through the point (3, 1). Its slope there must be

dy/dx = 3 − 2(1) = 1

So the curve crosses (3, 1) climbing at 45°. You can draw a short segment through (3, 1) with slope 1 to record that fact.

Do the same at many points and you get a slope field (also called a direction field). It is a picture of the differential equation, built on a finite grid of points in the plane.

• Each segment is a small piece of a tangent line. It shows the direction a solution curve would travel if it passed through that point.
• The segments do not show the solution itself. They are like wind arrows on a weather map: each one gives a direction, and a solution curve is a path that follows them.
• Because a solution curve must have the slope given by the equation at every point it passes through, the field shows how the whole family of solutions behaves. That is why a slope field is a way to estimate solutions without solving anything.

Notice what you need to make one: only substitution. No integration is involved.

What the angles look like

Draw every segment the same length and centred on its point. Only the tilt carries information.

Slope	How the segment looks
0	Horizontal
1/2	Gentle rise, about 27° above horizontal
1	Rises at 45°
2	Steeper, about 63°
3	Very steep, about 72°
−1	Falls at 45° (going left to right)

You do not need a protractor. Just keep the relative steepness consistent: a slope of 2 must look steeper than a slope of 1, and a slope of −1 must mirror a slope of 1. These angles assume the same scale on both axes.

How to sketch a slope field by hand

Questions usually give you the points, often a small grid such as nine or fifteen dots on a set of axes. Work in three steps.

1. Make a table. List the points and substitute each one into dy/dx. Keep exact values such as 1/2.
2. Look for structure first. Before drawing, see which rows, columns or lines share the same slope. This saves time and catches arithmetic slips.
3. Draw. At each point, draw a short segment centred on the point with the computed slope. Horizontal for 0, rising to the right for positive, falling to the right for negative, steeper for larger size.

The table is your working. If a question asks you to sketch a slope field, the segments are what is assessed, but the table is how you make sure they are right.

Worked example 1: drawing a slope field from a table

Question. Sketch the slope field for dy/dx = x(y − 1)/2 at the fifteen points with x = −2, −1, 0, 1, 2 and y = 0, 1, 2.

1. Look for structure.
   • If x = 0, then dy/dx = 0 for every y. So the whole column x = 0 is horizontal.
   • If y = 1, then y − 1 = 0, so dy/dx = 0 for every x. The whole row y = 1 is horizontal.
2. Make the table. Substitute each remaining point. For example, at (2, 0): dy/dx = 2(0 − 1)/2 = −1. At (−1, 2): dy/dx = (−1)(2 − 1)/2 = −1/2.

	x = −2	x = −1	x = 0	x = 1	x = 2
y = 2	−1	−1/2	0	1/2	1
y = 1	0	0	0	0	0
y = 0	1	1/2	0	−1/2	−1

3. Draw the segments. Each segment is centred on its point. In the row y = 2, the slopes go from falling at 45° on the left, through gentle falling, flat, gentle rising, to rising at 45° on the right. The row y = 0 is the mirror image of that pattern.

Check by signs. The sign of x(y − 1) tells you where segments rise and fall. Above y = 1 (so y − 1 > 0), the sign matches the sign of x: falling on the left, rising on the right. Below y = 1 it is the other way round. The table agrees.

Interpretation. Any solution curve that meets the line y = 1 meets it travelling horizontally, and the same is true where a solution crosses the y-axis.

Patterns that let you read a field quickly

Most fields you meet fall into one of three types. Recognising the type is the fastest way to match an equation to a picture.

If dy/dx depends on…	What you see	Example
x only	Every vertical column is identical: all segments directly above each other are parallel.	dy/dx = sin x: horizontal along x = 0 and x = π, slope 1 along x = π/2
y only	Every horizontal row is identical: all segments side by side are parallel.	dy/dx = y/2: horizontal along y = 0, slope 1 along y = 2, slope −1 along y = −2
x and y	Neither rows nor columns repeat. Look for the curve where dy/dx = 0.	dy/dx = x(y − 1)/2 from Worked example 1

Four more features to test:

• Zero slopes. Solve dy/dx = 0. The solutions are lines or curves in the plane along which every segment is horizontal. For dy/dx = x − 2y this is the line y = x/2.
• Signs. Where dy/dx > 0, segments rise to the right; where dy/dx < 0, they fall. The zero-slope curves usually separate these regions.
• Size. Where |dy/dx| is large, segments are steep. For dy/dx = 1 − y², the segments are flat along y = ±1 and get steeper the further you go above y = 1 or below y = −1.
• Test points. Pick one or two easy points, such as (0, 0) or (1, 0), and compare the computed slope with the picture. One clear mismatch rules an equation out.

Worked example 2: matching an equation to a field

Question. The slope field below is drawn at the points with x and y each going from −2 to 2 in steps of 0.5. Which differential equation does it show?

• (A) dy/dx = 1 − x²
• (B) dy/dx = 1 − y²
• (C) dy/dx = y² − 1
• (D) dy/dx = 1 − xy

1. Look at the rows and columns. Every segment in a horizontal row has the same slope; the columns are not identical. So dy/dx depends on y only. That rules out (A), which depends on x only, and (D), which depends on both.
2. Find the zero slopes. The segments are horizontal along y = 1 and y = −1. Both (B) and (C) are 0 there, so this does not decide it.
3. Use a test point. Along the x-axis (y = 0), the segments rise at 45°, so the slope is 1. Option (B) gives 1 − 0² = 1. Option (C) gives 0² − 1 = −1, which would fall.
4. Confirm with a second point. At y = 2 the segments fall steeply. Option (B) gives 1 − 4 = −3, a steep fall. That matches.

Answer. (B), dy/dx = 1 − y².

Why the steps are in this order. Structure (rows or columns) eliminates the most options for the least work. Zero slopes then narrow it further. A test point settles a choice between two equations that differ only in sign, which is a very common trap.

A first look at solution curves

Because every segment is a piece of a tangent line, you can trace an approximate solution curve: start at a point and move so that the curve is always tangent to the segments near it. The curve does not have to pass through the grid points; between them, blend the directions smoothly.

For dy/dx = 1 − y², the curve through (0, 0) rises steeply at first, then levels off as it approaches y = 1 on the right. Going left from (0, 0), it falls and levels off towards y = −1. (For background only: this particular solution is y = tanh x, the hyperbolic tangent, which is not part of the course. You do not need it to draw the curve.)

Topic 7.4 develops this into full reasoning: families of solutions, solutions that stay constant, and what happens in the long run.

Common misconceptions

• “The segments are the graph of the solution.” They are tangent directions. A solution curve follows them but is not made of them joined end to end.
• Swapping x and y. At (1, −2) the x-value is 1. Substituting the numbers in the wrong places is the most common arithmetic error in this topic.
• Long lines instead of short segments. A long line suggests the slope stays the same along it. Keep segments short and centred.
• Inconsistent steepness. If a slope of 2 looks the same as a slope of 1/2, the field misrepresents the equation. Compare segments with each other, not just with the horizontal.
• Leaving out the zero slopes. A slope of 0 still needs a segment: a horizontal one. A blank point is wrong.
• Rows and columns mixed up. dy/dx depending on x only makes identical columns (vertical lines of segments), not rows.
• Matching on zero slopes alone. dy/dx = 1 − y² and dy/dx = y² − 1 have the same horizontal segments but opposite slopes everywhere else. Always check a sign with a test point.
• Thinking a field gives exact values. It gives directions and shapes, so any value read from a sketched curve is an estimate.

Where this leads

In Topic 7.4 you will use slope fields to reason about solutions: sketch the particular solution through a given point, spot constant solutions, describe long-run behaviour, and connect the field with the sign of the second derivative. Later in the unit, separation of variables gives exact solutions you can check against the field. BC students will also meet Euler’s method, which follows the segments step by step to estimate values.

Next topic: Reasoning Using Slope Fields. Students in either course can return to the Calculus AB roadmap or the Calculus BC roadmap.

Try the practice questions now, then use the revision notes and the checklist to consolidate.