Unit 7Slope Fields

Concept 4.113Built for May 2027 AP Exam

Slope Fields

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

A slope field draws $\frac{dy}{dx}$ at many points at once. Each short dash is the tangent direction a solution curve would have if it passed through that point, so the picture encodes the differential equation without ever producing a function. To read it, start at the given point and move right, turning so that the curve stays parallel to the dashes it crosses; the result is the particular solution, sketched rather than calculated. Horizontal dashes mark where $\frac{dy}{dx} = 0$ , and a whole row of them means an equilibrium value that solutions approach but never cross. The field never tells you the scale of the axes or the value of a constant, which is why a sketch is qualitative: it carries shape and limiting behaviour, not numbers.

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Almost always a printed field with one part asking you to sketch a solution curve through a marked point, and often a second part asking what the long-run behaviour of that solution is. Recent exams also ask you to decide whether a given picture could be the field for a stated equation, and to explain the rejection.
What you’ll do: Identify the point, follow the dashes to the right and to the left, keep the curve tangent to every dash it touches, and mark any horizontal-dash line as a value the solution approaches rather than reaches.
Memory hook: The dashes are directions, not points. A curve that merely passes near a dash is wrong; it has to be tangent to it.
Use it on the exam: For MCQs: Compute $\frac{dy}{dx}$ at one or two convenient points and compare with the picture. One mismatched sign usually eliminates three options. For FRQs: Sketch with a smooth curve, not a polyline through the dashes, and say in words what the sketch shows — increasing, concave up, approaching a horizontal line. The description earns points the drawing alone does not.

Concept Review

What a Slope Field Is

Figure 1 — Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

A slope field is a grid of short segments, each drawn with the slope the differential equation assigns to that point. It is a picture of the right-hand side $F(x, y)$ , nothing more. Two facts follow immediately, and both are worth more than any amount of memorisation:

Every point in the plane has exactly one slope, so the dashes never contradict each other.
A solution curve is tangent to the dash at each of its own points, so the field constrains the whole shape of the solution while saying nothing about where it starts.

Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

Building One by Hand

For $\frac{dy}{dx} = \frac{x + y}{3}$ , substituting a few points gives the table below. The exam will never ask you to draw a full field by hand, but it will ask you to check whether a printed field matches an equation, and that check is exactly this table applied at one or two points.

Point $(x, y)$	$\frac{x+y}{3}$	Dash direction
$(-3, 1)$	$-\frac{2}{3}$	down to the right, moderately steep
$(-3, -1)$	$-\frac{4}{3}$	down to the right, steep
$(0, 0)$	$0$	horizontal
$(0, 1)$	$\frac{1}{3}$	up to the right, gentle
$(3, -1)$	$\frac{2}{3}$	up to the right, moderately steep
$(3, 1)$	$\frac{4}{3}$	up to the right, steep

The pattern in the table is the pattern in the picture: slopes increase as you move right and as you move up, because both $x$ and $y$ appear with positive coefficients.

Sketching a Solution Through a Point

Start at the marked point. Move a short distance to the right, choosing the direction that matches the local dash, then repeat. Do the same to the left. The result is a curve whose tangent at every point agrees with the field, which is precisely the definition of a solution.

Three habits make the sketch reliable:

Keep it smooth. A solution is differentiable, so the sketch has no corners even where the dashes change direction abruptly.
Stay tangent, not parallel-at-a-distance. The curve must match each dash at the point where it crosses that dash's location.
Note where it flattens. If the curve approaches a row of horizontal dashes, it flattens toward that value and never crosses it, because crossing would require the solution to meet another solution.

Reading Behaviour Without Solving

Several questions are answerable from the picture alone, and all of them score:

Where is the solution increasing? Wherever the dashes tilt up to the right.
Where is it concave up? Where the dashes tilt more steeply upward as you move right, or less steeply downward.
What is the long-run value? The horizontal-dash line the curve is drifting toward. If the dashes above it point down and the dashes below point up, that line is a stable equilibrium and every nearby solution converges to it.

For $\frac{dy}{dx} = \frac{x+y}{3}$ there is no horizontal-dash line except at isolated points, so no solution settles down; every solution eventually rises without bound. A field whose dashes are horizontal along an entire line, by contrast, has an equilibrium, and questions about carrying capacity or terminal temperature are really questions about that line.

What the Field Cannot Tell You

The field fixes shape, not position. Infinitely many solution curves are parallel copies of one another, and the initial condition chooses one. Two consequences show up on exams. First, a sketch without the marked point is not a particular solution and earns nothing. Second, the axes usually carry no numbers, so a sketch can show that a solution approaches $12$ only if the horizontal-dash line is labelled; otherwise the answer must be phrased qualitatively.

Common Misconception Check

Myth 1: “A solution curve is the curve that connects the dashes end to end.” Truth: It is tangent to the dashes, which is a different and stronger condition. Connecting ends produces a curve with the wrong slope between grid points.
Myth 2: “If two dashes point in different directions, the field is inconsistent.” Truth: Different points have different slopes. Only the same point having two slopes would be a contradiction.
Myth 3: “The sketch gives the value of the constant $C$ .” Truth: The constant comes from the initial condition. The field shows the family's shape, and the point selects one member.

Topic-Specific Study Method

Enter the Field, Then Turn With It

Borrowed idea: A slope field is a picture of the derivative, not of the solution. Each dash is short enough that its slope stands for the slope at a single point, so the field tells you the direction a solution must take without ever naming the function. Read it in one direction only: from left to right. Nothing here is integrated. A solution curve is the path that stays tangent to every dash it touches, which is why you sketch it by entering the field and turning with the marks instead of plotting points from a formula.

Exam disguise	What you see on the page	What your brain must translate	Required execution
Sketch through a point	a dot on a printed field	the dot is the initial condition	draw smooth and tangent, both directions
“Could this be the field for …”	a field plus a stated equation	test one point where signs differ	compute $\frac{dy}{dx}$ there and compare
Horizontal row of dashes	a line of flat dashes	an equilibrium value	describe convergence, never crossing
Behaviour question	“what is $\lim_{t\to\infty} y$ ”	read the attracting line	name it, or say it does not exist
Field with no numbers	unlabelled axes	only qualitative claims are available	phrase answers in words, not values

① Mark the given point before drawing anything. The point is the anchor, and a sketch that starts elsewhere answers a different question.

② Draw in short strokes, checking the tangent at each grid line you cross. Long sweeping strokes hide a wrong slope for several units.

③ Finish with one sentence of description: increasing or decreasing, concave up or down, approaching which line. That sentence is a scored component on its own.

Training & Error Diagnosis

Core Archetype (Simplest Form): A slope field for $\frac{dy}{dx} = x - y$ is given, with $y(0) = 1$ . Describe the solution curve.

Standard Execution:

Slope at the point. At $(0,1)$ , $\frac{dy}{dx} = -1$ , so the curve leaves the point heading down to the right at $45°$ .

Follow right. The dashes flatten as $y$ falls toward the line $y = x - 1$ , so the curve keeps decreasing but less steeply.

Follow left. Going left, $x - y$ becomes negative and small, so the curve rises gently to the left.

Describe. The solution decreases, is concave up, and approaches the line $y = x - 1$ from above without touching it.

Exam Variation (Context / Parameter Shift): A field is proposed for $\frac{dy}{dx} = y - 2$ , and in the picture every dash in the second quadrant tilts up to the right. Decide whether the picture is possible.

Solution: In the second quadrant $x < 0$ and $y > 0$ . Above the line $y = 2$ we get $y - 2 > 0$ , so dashes must tilt up to the right; below $y = 2$ they must tilt down. A picture showing every second-quadrant dash tilting up therefore forces $y > 2$ throughout that quadrant, which contradicts the quadrant containing points with $0 < y < 2$ . The correct reason to give is a single point, for instance $(0,1)$ , where the field must show slope $-1$ rather than a positive slope.

Common Traps & Failure Signals:

Polyline sketch (Signal: visible corners at grid lines): solutions are differentiable, so the sketch must be smooth.
Wrong anchor (Signal: curve does not pass through the dot): the marked point is the initial condition.
Numeric answer from an unlabelled field (Signal: a number like $12$ with no scale on the axes): only qualitative claims are supported.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Dashes Read as Points: You connected the ends of the dashes into a path, which produces a curve whose slope between grid points belongs to no solution at all.
Anchor Ignored: The sketch has to start at the marked point.
Size Confused With Shape: You read a number off an axis that carries no numbers, and the qualitative answer that would have scored was never written.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Six of the nine released exams since 2017 print a slope field and ask for a solution curve through a marked point, and two of those six attach a further part about long-run behaviour or about a picture that cannot be the field
10-Year Total Count: 6 of 9 Exams · 6 FRQ Parts; The slope-field part is always the opening part of a differential-equation question, so it is scored before any separation happens
FRQ Years Appeared: 2018 (Q6a), 2021 (Q6a), 2022 (Q5a), 2023 (Q3a), 2024 (Q3a), 2026 (Q3A) ---

Practice Examples

MCQ 1

One Point Decides It

The slope field for $\frac{dy}{dx} = x - y$ is drawn. At the point $(1, 2)$ , the tangent to the solution curve has slope

MCQ 2

Where the Dashes Go Flat

In the slope field for $\frac{dy}{dx} = y - 2$ , every dash on which line is horizontal?

MCQ 3

What a Flat Dash Implies

A solution curve is sketched through a point where the field's dash is horizontal. Which statement must be true at that point?

FRQ 4

Reading and Sketching a Field

A slope field for the differential equation $\frac{dy}{dx} = \frac{x + y}{3}$ is shown, with a marked point at $(0, -0.2)$ .

(a) Complete a table giving $\frac{dy}{dx}$ at the four points $(-3, 1)$ , $(0, 0)$ , $(0, 1)$ , and $(3, -1)$ .

(b) Describe the solution curve passing through $(0, -0.2)$ : state whether it is increasing or decreasing on each side of the point, its concavity, and its behaviour as $x \to \infty$ .

(c) Explain why two distinct solution curves of this equation can never intersect.

Cheatsheet

Core Reference

	Action	Failure if skipped
1	compute $\frac{dy}{dx}$ at the marked point	the sketch starts in the wrong direction
2	move right in short strokes, tangent to each dash	a polyline with corners
3	repeat to the left	half the solution is missing
4	find horizontal-dash lines	equilibria and limits are missed
5	describe the shape in words	the scored sentence is absent

Sentences That Score

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Start Topic Practice Next Topic: 7.3 Separation of Variables

Unit 7Slope Fields

Concept 4.113Built for May 2027 AP Exam

Slope Fields

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Almost always a printed field with one part asking you to sketch a solution curve through a marked point, and often a second part asking what the long-run behaviour of that solution is. Recent exams also ask you to decide whether a given picture could be the field for a stated equation, and to explain the rejection.
What you’ll do: Identify the point, follow the dashes to the right and to the left, keep the curve tangent to every dash it touches, and mark any horizontal-dash line as a value the solution approaches rather than reaches.
Memory hook: The dashes are directions, not points. A curve that merely passes near a dash is wrong; it has to be tangent to it.
Use it on the exam: For MCQs: Compute $\frac{dy}{dx}$ at one or two convenient points and compare with the picture. One mismatched sign usually eliminates three options. For FRQs: Sketch with a smooth curve, not a polyline through the dashes, and say in words what the sketch shows — increasing, concave up, approaching a horizontal line. The description earns points the drawing alone does not.

Concept Review

What a Slope Field Is

Figure 1 — Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

Every point in the plane has exactly one slope, so the dashes never contradict each other.
A solution curve is tangent to the dash at each of its own points, so the field constrains the whole shape of the solution while saying nothing about where it starts.

Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

Building One by Hand

Point $(x, y)$	$\frac{x+y}{3}$	Dash direction
$(-3, 1)$	$-\frac{2}{3}$	down to the right, moderately steep
$(-3, -1)$	$-\frac{4}{3}$	down to the right, steep
$(0, 0)$	$0$	horizontal
$(0, 1)$	$\frac{1}{3}$	up to the right, gentle
$(3, -1)$	$\frac{2}{3}$	up to the right, moderately steep
$(3, 1)$	$\frac{4}{3}$	up to the right, steep

The pattern in the table is the pattern in the picture: slopes increase as you move right and as you move up, because both $x$ and $y$ appear with positive coefficients.

Sketching a Solution Through a Point

Three habits make the sketch reliable:

Keep it smooth. A solution is differentiable, so the sketch has no corners even where the dashes change direction abruptly.
Stay tangent, not parallel-at-a-distance. The curve must match each dash at the point where it crosses that dash's location.
Note where it flattens. If the curve approaches a row of horizontal dashes, it flattens toward that value and never crosses it, because crossing would require the solution to meet another solution.

Reading Behaviour Without Solving

Several questions are answerable from the picture alone, and all of them score:

Where is the solution increasing? Wherever the dashes tilt up to the right.
Where is it concave up? Where the dashes tilt more steeply upward as you move right, or less steeply downward.
What is the long-run value? The horizontal-dash line the curve is drifting toward. If the dashes above it point down and the dashes below point up, that line is a stable equilibrium and every nearby solution converges to it.

What the Field Cannot Tell You

Common Misconception Check

Myth 1: “A solution curve is the curve that connects the dashes end to end.” Truth: It is tangent to the dashes, which is a different and stronger condition. Connecting ends produces a curve with the wrong slope between grid points.
Myth 2: “If two dashes point in different directions, the field is inconsistent.” Truth: Different points have different slopes. Only the same point having two slopes would be a contradiction.
Myth 3: “The sketch gives the value of the constant $C$ .” Truth: The constant comes from the initial condition. The field shows the family's shape, and the point selects one member.

Topic-Specific Study Method

Enter the Field, Then Turn With It

Exam disguise	What you see on the page	What your brain must translate	Required execution
Sketch through a point	a dot on a printed field	the dot is the initial condition	draw smooth and tangent, both directions
“Could this be the field for …”	a field plus a stated equation	test one point where signs differ	compute $\frac{dy}{dx}$ there and compare
Horizontal row of dashes	a line of flat dashes	an equilibrium value	describe convergence, never crossing
Behaviour question	“what is $\lim_{t\to\infty} y$ ”	read the attracting line	name it, or say it does not exist
Field with no numbers	unlabelled axes	only qualitative claims are available	phrase answers in words, not values

① Mark the given point before drawing anything. The point is the anchor, and a sketch that starts elsewhere answers a different question.

② Draw in short strokes, checking the tangent at each grid line you cross. Long sweeping strokes hide a wrong slope for several units.

③ Finish with one sentence of description: increasing or decreasing, concave up or down, approaching which line. That sentence is a scored component on its own.

Training & Error Diagnosis

Core Archetype (Simplest Form): A slope field for $\frac{dy}{dx} = x - y$ is given, with $y(0) = 1$ . Describe the solution curve.

Standard Execution:

Slope at the point. At $(0,1)$ , $\frac{dy}{dx} = -1$ , so the curve leaves the point heading down to the right at $45°$ .

Follow right. The dashes flatten as $y$ falls toward the line $y = x - 1$ , so the curve keeps decreasing but less steeply.

Follow left. Going left, $x - y$ becomes negative and small, so the curve rises gently to the left.

Describe. The solution decreases, is concave up, and approaches the line $y = x - 1$ from above without touching it.

Solution: In the second quadrant $x < 0$ and $y > 0$ . Above the line $y = 2$ we get $y - 2 > 0$ , so dashes must tilt up to the right; below $y = 2$ they must tilt down. A picture showing every second-quadrant dash tilting up therefore forces $y > 2$ throughout that quadrant, which contradicts the quadrant containing points with $0 < y < 2$ . The correct reason to give is a single point, for instance $(0,1)$ , where the field must show slope $-1$ rather than a positive slope.

Common Traps & Failure Signals:

Polyline sketch (Signal: visible corners at grid lines): solutions are differentiable, so the sketch must be smooth.
Wrong anchor (Signal: curve does not pass through the dot): the marked point is the initial condition.
Numeric answer from an unlabelled field (Signal: a number like $12$ with no scale on the axes): only qualitative claims are supported.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Dashes Read as Points: You connected the ends of the dashes into a path, which produces a curve whose slope between grid points belongs to no solution at all.
Anchor Ignored: The sketch has to start at the marked point.
Size Confused With Shape: You read a number off an axis that carries no numbers, and the qualitative answer that would have scored was never written.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Six of the nine released exams since 2017 print a slope field and ask for a solution curve through a marked point, and two of those six attach a further part about long-run behaviour or about a picture that cannot be the field
10-Year Total Count: 6 of 9 Exams · 6 FRQ Parts; The slope-field part is always the opening part of a differential-equation question, so it is scored before any separation happens
FRQ Years Appeared: 2018 (Q6a), 2021 (Q6a), 2022 (Q5a), 2023 (Q3a), 2024 (Q3a), 2026 (Q3A) ---

Practice Examples

MCQ 1

One Point Decides It

The slope field for $\frac{dy}{dx} = x - y$ is drawn. At the point $(1, 2)$ , the tangent to the solution curve has slope

MCQ 2

Where the Dashes Go Flat

In the slope field for $\frac{dy}{dx} = y - 2$ , every dash on which line is horizontal?

MCQ 3

What a Flat Dash Implies

A solution curve is sketched through a point where the field's dash is horizontal. Which statement must be true at that point?

FRQ 4

Reading and Sketching a Field

A slope field for the differential equation $\frac{dy}{dx} = \frac{x + y}{3}$ is shown, with a marked point at $(0, -0.2)$ .

(a) Complete a table giving $\frac{dy}{dx}$ at the four points $(-3, 1)$ , $(0, 0)$ , $(0, 1)$ , and $(3, -1)$ .

(b) Describe the solution curve passing through $(0, -0.2)$ : state whether it is increasing or decreasing on each side of the point, its concavity, and its behaviour as $x \to \infty$ .

(c) Explain why two distinct solution curves of this equation can never intersect.

Cheatsheet

Core Reference

	Action	Failure if skipped
1	compute $\frac{dy}{dx}$ at the marked point	the sketch starts in the wrong direction
2	move right in short strokes, tangent to each dash	a polyline with corners
3	repeat to the left	half the solution is missing
4	find horizontal-dash lines	equilibria and limits are missed
5	describe the shape in words	the scored sentence is absent

Sentences That Score

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 7.3 Separation of Variables

Unit 7Slope Fields

Concept 4.113Built for May 2027 AP Exam

Slope Fields

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus AB tests this skill through the representations and reasoning described below.

What you’ll see: Almost always a printed field with one part asking you to sketch a solution curve through a marked point, and often a second part asking what the long-run behaviour of that solution is. Recent exams also ask you to decide whether a given picture could be the field for a stated equation, and to explain the rejection.
What you’ll do: Identify the point, follow the dashes to the right and to the left, keep the curve tangent to every dash it touches, and mark any horizontal-dash line as a value the solution approaches rather than reaches.
Memory hook: The dashes are directions, not points. A curve that merely passes near a dash is wrong; it has to be tangent to it.
Use it on the exam: For MCQs: Compute $\frac{dy}{dx}$ at one or two convenient points and compare with the picture. One mismatched sign usually eliminates three options. For FRQs: Sketch with a smooth curve, not a polyline through the dashes, and say in words what the sketch shows — increasing, concave up, approaching a horizontal line. The description earns points the drawing alone does not.

Concept Review

What a Slope Field Is

Figure 1 — Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

Every point in the plane has exactly one slope, so the dashes never contradict each other.
A solution curve is tangent to the dash at each of its own points, so the field constrains the whole shape of the solution while saying nothing about where it starts.

Every dash is the value of $\frac{dy}{dx}$ at its own point. The blue curve is the one solution that passes through the marked point.

Building One by Hand

Point $(x, y)$	$\frac{x+y}{3}$	Dash direction
$(-3, 1)$	$-\frac{2}{3}$	down to the right, moderately steep
$(-3, -1)$	$-\frac{4}{3}$	down to the right, steep
$(0, 0)$	$0$	horizontal
$(0, 1)$	$\frac{1}{3}$	up to the right, gentle
$(3, -1)$	$\frac{2}{3}$	up to the right, moderately steep
$(3, 1)$	$\frac{4}{3}$	up to the right, steep

The pattern in the table is the pattern in the picture: slopes increase as you move right and as you move up, because both $x$ and $y$ appear with positive coefficients.

Sketching a Solution Through a Point

Three habits make the sketch reliable:

Keep it smooth. A solution is differentiable, so the sketch has no corners even where the dashes change direction abruptly.
Stay tangent, not parallel-at-a-distance. The curve must match each dash at the point where it crosses that dash's location.
Note where it flattens. If the curve approaches a row of horizontal dashes, it flattens toward that value and never crosses it, because crossing would require the solution to meet another solution.

Reading Behaviour Without Solving

Several questions are answerable from the picture alone, and all of them score:

Where is the solution increasing? Wherever the dashes tilt up to the right.
Where is it concave up? Where the dashes tilt more steeply upward as you move right, or less steeply downward.
What is the long-run value? The horizontal-dash line the curve is drifting toward. If the dashes above it point down and the dashes below point up, that line is a stable equilibrium and every nearby solution converges to it.

What the Field Cannot Tell You

Common Misconception Check

Myth 1: “A solution curve is the curve that connects the dashes end to end.” Truth: It is tangent to the dashes, which is a different and stronger condition. Connecting ends produces a curve with the wrong slope between grid points.
Myth 2: “If two dashes point in different directions, the field is inconsistent.” Truth: Different points have different slopes. Only the same point having two slopes would be a contradiction.
Myth 3: “The sketch gives the value of the constant $C$ .” Truth: The constant comes from the initial condition. The field shows the family's shape, and the point selects one member.

Topic-Specific Study Method

Enter the Field, Then Turn With It

Exam disguise	What you see on the page	What your brain must translate	Required execution
Sketch through a point	a dot on a printed field	the dot is the initial condition	draw smooth and tangent, both directions
“Could this be the field for …”	a field plus a stated equation	test one point where signs differ	compute $\frac{dy}{dx}$ there and compare
Horizontal row of dashes	a line of flat dashes	an equilibrium value	describe convergence, never crossing
Behaviour question	“what is $\lim_{t\to\infty} y$ ”	read the attracting line	name it, or say it does not exist
Field with no numbers	unlabelled axes	only qualitative claims are available	phrase answers in words, not values

① Mark the given point before drawing anything. The point is the anchor, and a sketch that starts elsewhere answers a different question.

② Draw in short strokes, checking the tangent at each grid line you cross. Long sweeping strokes hide a wrong slope for several units.

③ Finish with one sentence of description: increasing or decreasing, concave up or down, approaching which line. That sentence is a scored component on its own.

Training & Error Diagnosis

Core Archetype (Simplest Form): A slope field for $\frac{dy}{dx} = x - y$ is given, with $y(0) = 1$ . Describe the solution curve.

Standard Execution:

Slope at the point. At $(0,1)$ , $\frac{dy}{dx} = -1$ , so the curve leaves the point heading down to the right at $45°$ .

Follow right. The dashes flatten as $y$ falls toward the line $y = x - 1$ , so the curve keeps decreasing but less steeply.

Follow left. Going left, $x - y$ becomes negative and small, so the curve rises gently to the left.

Describe. The solution decreases, is concave up, and approaches the line $y = x - 1$ from above without touching it.

Solution: In the second quadrant $x < 0$ and $y > 0$ . Above the line $y = 2$ we get $y - 2 > 0$ , so dashes must tilt up to the right; below $y = 2$ they must tilt down. A picture showing every second-quadrant dash tilting up therefore forces $y > 2$ throughout that quadrant, which contradicts the quadrant containing points with $0 < y < 2$ . The correct reason to give is a single point, for instance $(0,1)$ , where the field must show slope $-1$ rather than a positive slope.

Common Traps & Failure Signals:

Polyline sketch (Signal: visible corners at grid lines): solutions are differentiable, so the sketch must be smooth.
Wrong anchor (Signal: curve does not pass through the dot): the marked point is the initial condition.
Numeric answer from an unlabelled field (Signal: a number like $12$ with no scale on the axes): only qualitative claims are supported.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Dashes Read as Points: You connected the ends of the dashes into a path, which produces a curve whose slope between grid points belongs to no solution at all.
Anchor Ignored: The sketch has to start at the marked point.
Size Confused With Shape: You read a number off an axis that carries no numbers, and the qualitative answer that would have scored was never written.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; Six of the nine released exams since 2017 print a slope field and ask for a solution curve through a marked point, and two of those six attach a further part about long-run behaviour or about a picture that cannot be the field
10-Year Total Count: 6 of 9 Exams · 6 FRQ Parts; The slope-field part is always the opening part of a differential-equation question, so it is scored before any separation happens
FRQ Years Appeared: 2018 (Q6a), 2021 (Q6a), 2022 (Q5a), 2023 (Q3a), 2024 (Q3a), 2026 (Q3A) ---

Practice Examples

MCQ 1

One Point Decides It

The slope field for $\frac{dy}{dx} = x - y$ is drawn. At the point $(1, 2)$ , the tangent to the solution curve has slope

MCQ 2

Where the Dashes Go Flat

In the slope field for $\frac{dy}{dx} = y - 2$ , every dash on which line is horizontal?

MCQ 3

What a Flat Dash Implies

A solution curve is sketched through a point where the field's dash is horizontal. Which statement must be true at that point?

FRQ 4

Reading and Sketching a Field

A slope field for the differential equation $\frac{dy}{dx} = \frac{x + y}{3}$ is shown, with a marked point at $(0, -0.2)$ .

(a) Complete a table giving $\frac{dy}{dx}$ at the four points $(-3, 1)$ , $(0, 0)$ , $(0, 1)$ , and $(3, -1)$ .

(b) Describe the solution curve passing through $(0, -0.2)$ : state whether it is increasing or decreasing on each side of the point, its concavity, and its behaviour as $x \to \infty$ .

(c) Explain why two distinct solution curves of this equation can never intersect.

Cheatsheet

Core Reference

	Action	Failure if skipped
1	compute $\frac{dy}{dx}$ at the marked point	the sketch starts in the wrong direction
2	move right in short strokes, tangent to each dash	a polyline with corners
3	repeat to the left	half the solution is missing
4	find horizontal-dash lines	equilibria and limits are missed
5	describe the shape in words	the scored sentence is absent

Sentences That Score

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