Reasoning Using Slope Fields
Reason about a solution family, equilibrium, direction, and concavity from a slope field and equation.
College Board Topic 7.4 · CED p. 135 (PDF p. 140) · CED effective Fall 2020; project course outline 2026–27.
A solution curve follows the field; an initial condition picks its starting point. Sketch a smooth curve tangent to nearby segments, and infer where it increases, decreases, or levels off. For y′=2−y, the horizontal line y=2 is an equilibrium solution. Below 2, slopes are positive; above 2, slopes are negative. Solutions move toward 2 as x increases.
A slope field represents a family of possible functions, not one answer. To infer concavity, differentiate the differential equation using the ordinary product or chain rule as needed. For y′=2−y, differentiating both sides gives y″=−y′, so a rising solution below 2 is concave down. In another example, y′=xy gives y″=y+xy′ by the product rule because y depends on x. A sketch is an estimate; do not report an exact coordinate from a coarse field unless the model gives it.
Let's work through it
- For y′=2−y with y(0)=1, the initial slope is 1. Reason: Use the initial point in the field rule.
- The curve increases while below 2 and bends downward because y″=−y′<0. Reason: The equation gives both direction and concavity.
- It approaches the equilibrium y=2 in the forward direction. Reason: The smooth model has positive slopes below 2 that shrink toward zero.
A zero-slope location is not necessarily an equilibrium
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Try it from memory
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Question 1
For y′=2−y starting above 2, does y increase or decrease?
See every recall question with its answer
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Retrieval 1
For y′=2−y starting above 2, does y increase or decrease?
Worked answer and reason
Decrease.
The slope is negative when y>2.
Retrieval 2
Is y=2 a solution of y′=2−y?
Worked answer and reason
Yes.
A constant has derivative zero, and the right side is zero at y=2.
Catch the tempting mistake
- A zero slope at one point means the whole solution is constant.
- An equilibrium requires the constant function to satisfy the equation for all relevant x, not just one point.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
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- Unit 7 problem 9 · No calculator · graphical
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- Unit 7 problem 14 · No calculator · analytic
Explain and apply the mathematics
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Follow a turning curve through a two-coordinate field
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For the teacher
- Ask students to explain their sketch with derivative signs and concavity. Use the same equation at two different initial heights to show a family of solutions.
Glossary and deeper reading · 8 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Initial condition
An initial condition specifies a solution’s value at a particular input. It can determine the constant in a family of solutions to a differential equation. For example, y′=2x with y(0)=3 selects y=x²+3. The solution must satisfy both the equation and the stated condition on a suitable interval.
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Slope field
A slope field displays short line segments with slopes specified by a first-order differential equation. For y′=F(x,y), the segment at (x,y) has slope F(x,y). For example, y′=y gives horizontal segments along y=0 and positive slopes above that axis. Solution curves follow the local slopes, and an initial point helps choose the relevant curve.
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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Local extremum
A local extremum is a value at least as large as, or at least as small as, all values at nearby allowed inputs. A local maximum compares nearby heights above, while a local minimum compares them below. For example, x² has a local minimum at 0. A local extremum need not be the greatest or least value over the entire domain.
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Differential equation
A differential equation relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true throughout an appropriate interval. For example, y′=2x has solutions y=x²+C. An initial condition can select a particular member of a family of solutions.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.