Unit 4: Functions Involving Parameters, Vectors & Matrices

Implicit Functions & Conic Sections

Representing conic sections and other implicitly defined relations.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Parametric Curves

Curves That Resist Being Solved for y

A circle's equation, x² + y² = 9, was never solved for y in the last lesson — it didn't need to be, to identify the shape and check points on it. Before reading on: are there curves where solving for y wouldn't just be unnecessary, but would actually make the equation harder to work with?

Definition — Implicit Relation

An implicit relation is an equation in x and y that isn't solved for either variable. Many implicit relations, including most conic sections and circles, aren't functions at all — a vertical line can cross their graph more than once — but individual points can still be verified by direct substitution, and features like symmetry, intercepts, and shape can be analyzed without ever isolating y.

Worked Example — Classifying a Conic from Its Implicit Equation

Classify 4x² + 9y² = 36. Dividing by 36 gives x²/9 + y²/4 = 1 — an ellipse, since both squared terms are positive with different denominators. Here a² = 9 (a = 3, the longer axis along x) and b² = 4 (b = 2, along y). The foci satisfy c² = a² − b² = 9 − 4 = 5, so c = √5, placing the foci at (±√5, 0). None of this required solving the equation for y.

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Worked Example — Verifying a Point on an Implicit, Non-Function Curve

The curve x³ + y³ − 3xy = 0 has no clean, elementary form solved for y — but a specific point can still be checked directly. Test (3/2, 3/2): (3/2)³ + (3/2)³ − 3(3/2)(3/2) = 27/8 + 27/8 − 27/4 = 54/8 − 54/8 = 0, confirming the point lies on the curve. Compare to the circle x² + y² = 25: testing (3, 4) gives 9 + 16 = 25, confirmed the same direct way. Both curves were verified by substitution alone, without ever needing y = (some expression in x).

Tip

Before attempting to solve an implicit relation for y, ask whether the question actually requires it — checking a point, finding intercepts (set the other variable to 0), and identifying a conic's type from its coefficients are all implicit-form tasks that solving for y wouldn't make any easier.

Common Mistakes

Assuming every equation in x and y can be rearranged into a single y = f(x) form.
A circle already needs two branches (y = ±√(...)) just to become functions, and curves like x³ + y³ − 3xy = 0 resist a clean rearrangement altogether — treating an implicit relation as a disguised function isn't always possible.
Confusing an ellipse's c² = a² − b² relationship with a hyperbola's c² = a² + b².
An ellipse's foci sit inside the curve (c < a, so c² = a² − b²); a hyperbola's foci sit outside, beyond the vertices (c > a, so c² = a² + b²) — the sign difference reflects that geometric distinction, not an arbitrary formula to memorize separately for each shape.

Key Takeaways

An implicit relation isn't solved for either variable, and often isn't a function of x — but points on it can still be verified by direct substitution.
A conic's type and key measurements (axes, foci) can be read from its implicit equation's coefficients without solving for y.
Some implicit relations, like x³ + y³ − 3xy = 0, have no clean elementary y = f(x) form at all — leaving them implicit is the practical choice, not a shortcut.

Summary

Implicit relations round out this course's representations: explicit, parametric, and implicit descriptions of the same kinds of curves. The next lesson turns from curves to a different mathematical object entirely — vectors, and the functions built from them.

Track your progress

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