Slope Fields

You’ll be able to

Interpret a slope field as a picture of a differential equation
Sketch a solution curve through a given point on a slope field
Match a differential equation to its slope field

A differential equation gives slopes everywhere

A differential equation like dy/dx = f(x, y) does not hand you a function directly — it tells you the slope of the solution at every point (x, y) in the plane. A slope field (or direction field) visualizes this by drawing a short line segment at a grid of points, each with the slope the equation prescribes there. The field is a "flow" that solution curves must follow: a solution is any curve that is tangent to the segments everywhere it passes.

Reading the field

To sketch a solution through a given point, start there and draw a curve that stays parallel to the nearby segments, letting the slopes steer you left and right. Where dy/dx = 0 the segments are horizontal (possible extrema); where dy/dx is large the segments are steep. If the slope depends only on x, every column looks the same; if it depends only on y, every row looks the same. Those symmetries are the fastest way to match an equation to its field.

What a slope field encodes

At each point (x, y), segment slope = dy/dx = f(x, y)

The field is the geometry of the equation; a solution curve threads through it tangent to every segment.

Worked example

For the differential equation dy/dx = x, describe the slope field and the shape of its solution curves.

1.The slope depends only on x, so every point in a given vertical column has the same slope — the field is identical up each column.
2.At x = 0 the segments are horizontal (slope 0); for x > 0 they tilt up, for x < 0 they tilt down.
3.Solutions have dy/dx = x, so antidifferentiating gives y = x²/2 + C — a family of upward parabolas.
4.Each parabola is a vertical shift of the others, matching the column-by-column repetition of the field.

Answer: The field repeats across each column (slope depends only on x), and the solution curves are the parabolas y = x²/2 + C.

Tip

To match an equation to a slope field quickly, test special points. Find where the slope is 0 (horizontal segments) and where it is undefined or steep. Those landmark locations usually eliminate every wrong choice.

Checkpoint

In a slope field for dy/dx = f(x, y), what does each small segment represent?

Checkpoint

For dy/dx = y, at every point where y = 2 the slope segments are:

On the exam

When a free-response question shows a slope field, you can sketch a particular solution without solving the equation — just start at the given point and follow the segments. Full credit comes from a curve tangent to the field, passing through the initial condition.

Answer the 2 checkpoints as you read.