Sketch the slope field for dy/dx = xy at the points shown. — points where the slope field is to be drawn, labeled -2, -1, 0, 1, 2, -1, 0, 1

x = -2 x = -1 x = 0 x = 1 x = 2; y = 1 -2 -1 0 1 2; y = 0 0 0 0 0 0; y = -1 2 1 0 -1 -2

On the slope field for dy/dx = y, sketch the solution through (0, 3) — a slope field

The curve through (0, 3) that follows the dashes

Sketch the slope field for dy/dx = -xy at the points shown. — points where the slope field is to be drawn, labeled -2, -1, 0, 1, 2, -1, 0, 1

x = -2 x = -1 x = 0 x = 1 x = 2; y = 1 2 1 0 -1 -2; y = 0 0 0 0 0 0; y = -1 -2 -1 0 1 2

On the slope field for dy/dx = x, sketch the solution through (0, 2) — a slope field

The curve through (0, 2) that follows the dashes

Sketch the slope field for dy/dx = 2x - y at the points shown. — points where the slope field is to be drawn, labeled -2, -1, 0, 1, 2, -1, 0, 1

x = -2 x = -1 x = 0 x = 1 x = 2; y = 1 -5 -3 -1 1 3; y = 0 -4 -2 0 2 4; y = -1 -3 -1 1 3 5

On the slope field for dy/dx = 1 - y, sketch the solution through (0, 3) — a slope field

The curve through (0, 3) that follows the dashes

Sketch the slope field for dy/dx = y^2 - x at the points shown. — points where the slope field is to be drawn, labeled -2, -1, 0, 1, 2, -1, 0, 1

x = -2 x = -1 x = 0 x = 1 x = 2; y = 1 3 2 1 0 -1; y = 0 2 1 0 -1 -2; y = -1 3 2 1 0 -1

On the slope field for dy/dx = y, sketch the solution through (0, -1) — a slope field

The curve through (0, -1) that follows the dashes

Sketch the slope field for dy/dx = 1 - y at the points shown. — points where the slope field is to be drawn, labeled -2, -1, 0, 1, 2, -1, 0, 1

x = -2 x = -1 x = 0 x = 1 x = 2; y = 1 0 0 0 0 0; y = 0 1 1 1 1 1; y = -1 2 2 2 2 2

On the slope field for dy/dx = x, sketch the solution through (0, -1) — a slope field

The curve through (0, -1) that follows the dashes

Sketch the slope field for dy/dx = -y at the points shown. — points where the slope field is to be drawn, labeled -1, 0, 1, -1, 0, 1

x = -1 x = 0 x = 1; y = 1 -1 -1 -1; y = 0 0 0 0; y = -1 1 1 1

On the slope field for dy/dx = -y, sketch the solution through (0, -1) — a slope field

The curve through (0, -1) that follows the dashes

Worksheets · Calculus

Slope fields worksheet

A slope field draws a short dash at each point, tilted to the slope dy/dx has there. To match one to its equation, check a few telling places: where the dashes are flat, whether dashes in a column all match (the equation has no y) or dashes in a row all match (no x), and what happens to the steepness as you move away from the axes.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Slope Fields Date Period Answer each question. Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = x - y & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = y - 1 & (D) \frac{d y}{d x} = 1 \end{array}$ For $\frac{d y}{d x} = 2 (y - 1)$ , find the slope of the field at $(0, - 3), (- 1, - 2), (- 2, 0)$ Sketch the slope field for $\frac{d y}{d x} = x y$ at the points shown. On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, 3)$ Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = y & (D) \frac{d y}{d x} = x \end{array}$ For $\frac{d y}{d x} = 3 (x + y)$ , find the slope of the field at $(- 2, 2), (0, 2), (3, - 2)$ Sketch the slope field for $\frac{d y}{d x} = - x y$ at the points shown. On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, 2)$ Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = y & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = - y & (D) \frac{d y}{d x} = y - x \end{array}$ For $\frac{d y}{d x} = - 3 y$ , find the slope of the field at $(2, - 1), (- 1, 1), (3, 3)$ Sketch the slope field for $\frac{d y}{d x} = 2 x - y$ at the points shown. On the slope field for $\frac{d y}{d x} = 1 - y$ , sketch the solution through $(0, 3)$ Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = - x & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = x + 1 & (D) \frac{d y}{d x} = 2 x \end{array}$ For $\frac{d y}{d x} = y - x$ , find the slope of the field at $(1, 3), (- 3, 3), (- 2, - 1)$ Sketch the slope field for $\frac{d y}{d x} = y^{2} - x$ at the points shown. On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, - 1)$ Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = - x y \\ (C) \frac{d y}{d x} = - x & (D) \frac{d y}{d x} = 1 \end{array}$ For $\frac{d y}{d x} = 2 (x - y)$ , find the slope of the field at $(- 1, - 3), (- 3, - 3), (- 1, 1)$ Sketch the slope field for $\frac{d y}{d x} = 1 - y$ at the points shown. On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, - 1)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Slope Fields

Date Period

Answer each question.

Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = x - y & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = y - 1 & (D) \frac{d y}{d x} = 1 \end{array}$
For $\frac{d y}{d x} = 2 (y - 1)$ , find the slope of the field at $(0, - 3), (- 1, - 2), (- 2, 0)$
Sketch the slope field for $\frac{d y}{d x} = x y$ at the points shown.
On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, 3)$
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = y & (D) \frac{d y}{d x} = x \end{array}$
For $\frac{d y}{d x} = 3 (x + y)$ , find the slope of the field at $(- 2, 2), (0, 2), (3, - 2)$
Sketch the slope field for $\frac{d y}{d x} = - x y$ at the points shown.
On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, 2)$
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = y & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = - y & (D) \frac{d y}{d x} = y - x \end{array}$
For $\frac{d y}{d x} = - 3 y$ , find the slope of the field at $(2, - 1), (- 1, 1), (3, 3)$
Sketch the slope field for $\frac{d y}{d x} = 2 x - y$ at the points shown.
On the slope field for $\frac{d y}{d x} = 1 - y$ , sketch the solution through $(0, 3)$
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = - x & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = x + 1 & (D) \frac{d y}{d x} = 2 x \end{array}$
For $\frac{d y}{d x} = y - x$ , find the slope of the field at $(1, 3), (- 3, 3), (- 2, - 1)$
Sketch the slope field for $\frac{d y}{d x} = y^{2} - x$ at the points shown.
On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, - 1)$
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = - x y \\ (C) \frac{d y}{d x} = - x & (D) \frac{d y}{d x} = 1 \end{array}$
For $\frac{d y}{d x} = 2 (x - y)$ , find the slope of the field at $(- 1, - 3), (- 3, - 3), (- 1, 1)$
Sketch the slope field for $\frac{d y}{d x} = 1 - y$ at the points shown.
On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, - 1)$



Sigma Prep · sigmaprep.io/worksheets Name Slope FieldsAnswer keyVersion 1 Date Period Answer each question. Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = x - y & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = y - 1 & (D) \frac{d y}{d x} = 1 \end{array}$ (A) For $\frac{d y}{d x} = 2 (y - 1)$ , find the slope of the field at $(0, - 3), (- 1, - 2), (- 2, 0)$ $- 8, - 6, - 2$ Sketch the slope field for $\frac{d y}{d x} = x y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & - 2 & - 1 & 0 & 1 & 2 \\ y = 0 & 0 & 0 & 0 & 0 & 0 \\ y = - 1 & 2 & 1 & 0 & - 1 & - 2 \end{array}$ On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, 3)$ The curve through $(0, 3)$ that follows the dashes Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = y & (D) \frac{d y}{d x} = x \end{array}$ (A) For $\frac{d y}{d x} = 3 (x + y)$ , find the slope of the field at $(- 2, 2), (0, 2), (3, - 2)$ $0, 6, 3$ Sketch the slope field for $\frac{d y}{d x} = - x y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 2 & 1 & 0 & - 1 & - 2 \\ y = 0 & 0 & 0 & 0 & 0 & 0 \\ y = - 1 & - 2 & - 1 & 0 & 1 & 2 \end{array}$ On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, 2)$ The curve through $(0, 2)$ that follows the dashes Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = y & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = - y & (D) \frac{d y}{d x} = y - x \end{array}$ (B) For $\frac{d y}{d x} = - 3 y$ , find the slope of the field at $(2, - 1), (- 1, 1), (3, 3)$ $3, - 3, - 9$ Sketch the slope field for $\frac{d y}{d x} = 2 x - y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & - 5 & - 3 & - 1 & 1 & 3 \\ y = 0 & - 4 & - 2 & 0 & 2 & 4 \\ y = - 1 & - 3 & - 1 & 1 & 3 & 5 \end{array}$ On the slope field for $\frac{d y}{d x} = 1 - y$ , sketch the solution through $(0, 3)$ The curve through $(0, 3)$ that follows the dashes Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = - x & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = x + 1 & (D) \frac{d y}{d x} = 2 x \end{array}$ (A) For $\frac{d y}{d x} = y - x$ , find the slope of the field at $(1, 3), (- 3, 3), (- 2, - 1)$ $2, 6, 1$ Sketch the slope field for $\frac{d y}{d x} = y^{2} - x$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 3 & 2 & 1 & 0 & - 1 \\ y = 0 & 2 & 1 & 0 & - 1 & - 2 \\ y = - 1 & 3 & 2 & 1 & 0 & - 1 \end{array}$ On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, - 1)$ The curve through $(0, - 1)$ that follows the dashes Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = - x y \\ (C) \frac{d y}{d x} = - x & (D) \frac{d y}{d x} = 1 \end{array}$ (D) For $\frac{d y}{d x} = 2 (x - y)$ , find the slope of the field at $(- 1, - 3), (- 3, - 3), (- 1, 1)$ $4, 0, - 4$ Sketch the slope field for $\frac{d y}{d x} = 1 - y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 0 & 0 & 0 & 0 & 0 \\ y = 0 & 1 & 1 & 1 & 1 & 1 \\ y = - 1 & 2 & 2 & 2 & 2 & 2 \end{array}$ On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, - 1)$ The curve through $(0, - 1)$ that follows the dashes Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Slope FieldsAnswer keyVersion 1

Date Period

Answer each question.

Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = x - y & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = y - 1 & (D) \frac{d y}{d x} = 1 \end{array}$ (A)
For $\frac{d y}{d x} = 2 (y - 1)$ , find the slope of the field at $(0, - 3), (- 1, - 2), (- 2, 0)$ $- 8, - 6, - 2$
Sketch the slope field for $\frac{d y}{d x} = x y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & - 2 & - 1 & 0 & 1 & 2 \\ y = 0 & 0 & 0 & 0 & 0 & 0 \\ y = - 1 & 2 & 1 & 0 & - 1 & - 2 \end{array}$
On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, 3)$ The curve through $(0, 3)$ that follows the dashes
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = y & (D) \frac{d y}{d x} = x \end{array}$ (A)
For $\frac{d y}{d x} = 3 (x + y)$ , find the slope of the field at $(- 2, 2), (0, 2), (3, - 2)$ $0, 6, 3$
Sketch the slope field for $\frac{d y}{d x} = - x y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 2 & 1 & 0 & - 1 & - 2 \\ y = 0 & 0 & 0 & 0 & 0 & 0 \\ y = - 1 & - 2 & - 1 & 0 & 1 & 2 \end{array}$
On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, 2)$ The curve through $(0, 2)$ that follows the dashes
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = y & (B) \frac{d y}{d x} = y^{2} \\ (C) \frac{d y}{d x} = - y & (D) \frac{d y}{d x} = y - x \end{array}$ (B)
For $\frac{d y}{d x} = - 3 y$ , find the slope of the field at $(2, - 1), (- 1, 1), (3, 3)$ $3, - 3, - 9$
Sketch the slope field for $\frac{d y}{d x} = 2 x - y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & - 5 & - 3 & - 1 & 1 & 3 \\ y = 0 & - 4 & - 2 & 0 & 2 & 4 \\ y = - 1 & - 3 & - 1 & 1 & 3 & 5 \end{array}$
On the slope field for $\frac{d y}{d x} = 1 - y$ , sketch the solution through $(0, 3)$ The curve through $(0, 3)$ that follows the dashes
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = - x & (B) \frac{d y}{d x} = x \\ (C) \frac{d y}{d x} = x + 1 & (D) \frac{d y}{d x} = 2 x \end{array}$ (A)
For $\frac{d y}{d x} = y - x$ , find the slope of the field at $(1, 3), (- 3, 3), (- 2, - 1)$ $2, 6, 1$
Sketch the slope field for $\frac{d y}{d x} = y^{2} - x$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 3 & 2 & 1 & 0 & - 1 \\ y = 0 & 2 & 1 & 0 & - 1 & - 2 \\ y = - 1 & 3 & 2 & 1 & 0 & - 1 \end{array}$
On the slope field for $\frac{d y}{d x} = y$ , sketch the solution through $(0, - 1)$ The curve through $(0, - 1)$ that follows the dashes
Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = - x y \\ (C) \frac{d y}{d x} = - x & (D) \frac{d y}{d x} = 1 \end{array}$ (D)
For $\frac{d y}{d x} = 2 (x - y)$ , find the slope of the field at $(- 1, - 3), (- 3, - 3), (- 1, 1)$ $4, 0, - 4$
Sketch the slope field for $\frac{d y}{d x} = 1 - y$ at the points shown. $\begin{array}{cccccc} x = - 2 & x = - 1 & x = 0 & x = 1 & x = 2 \\ y = 1 & 0 & 0 & 0 & 0 & 0 \\ y = 0 & 1 & 1 & 1 & 1 & 1 \\ y = - 1 & 2 & 2 & 2 & 2 & 2 \end{array}$
On the slope field for $\frac{d y}{d x} = x$ , sketch the solution through $(0, - 1)$ The curve through $(0, - 1)$ that follows the dashes

How to do these

Which has flat dashes all along the line $y = x$ : $\frac{d y}{d x} = x - y$ or $\frac{d y}{d x} = x + y$ ?

On the line y = x, x - y is 0 at every point, so every dash there is flat.
x + y is 0 on the line y = -x instead.

The answer is $\frac{d y}{d x} = x - y$ .

Where students go wrong

Reading the dashes as the solution curve. Each dash is only the direction at its own point; a solution follows the dashes from point to point and is a separate curve.

Also called slope fields, direction fields, matching slope fields or slope field AP calculus.

What you can put on this worksheet

Match the equation to its slope field: Which equation has this slope field? $\begin{array}{ll} (A) \frac{d y}{d x} = 2 x & (B) \frac{d y}{d x} = - y \\ (C) \frac{d y}{d x} = x^{2} & (D) \frac{d y}{d x} = x \end{array}$ → (A)
The slope of the field at a point: For $\frac{d y}{d x} = 2 x$ , find the slope of the field at $(0, 2), (- 3, 0), (0, - 2)$ → $0, - 6, 0$
Sketch it at given points: Sketch the slope field for $\frac{d y}{d x} = - y$ at the points shown. → $\begin{array}{cccc} x = - 1 & x = 0 & x = 1 \\ y = 1 & - 1 & - 1 & - 1 \\ y = 0 & 0 & 0 & 0 \\ y = - 1 & 1 & 1 & 1 \end{array}$
Sketch a solution curve: On the slope field for $\frac{d y}{d x} = - y$ , sketch the solution through $(0, - 1)$ → The curve through $(0, - 1)$ that follows the dashes

Questions about these worksheets

Yes. Every slope fields sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the slope fields questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: match the equation to its slope field, the slope of the field at a point, sketch it at given points and sketch a solution curve. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the dashes as the solution curve. Each dash is only the direction at its own point; a solution follows the dashes from point to point and is a separate curve.

Slope fields worksheet

Select your difficulty

Slope Fields

Slope FieldsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these slope fields worksheets free?+

Do the slope fields worksheets come with an answer key?+

Are these slope fields worksheets printable?+

What grade are these slope fields worksheets for?+

Can I get a different version of the same slope fields worksheet?+

Is this a free alternative to Kuta Software for slope fields?+

Can I choose how hard the slope fields questions are?+

What kinds of question are on a slope fields worksheet?+

What do students get wrong with slope fields?+