Conic Sections & Introduction to Limits

Parabolas & Ellipses

Graphing parabolas and ellipses from their standard-form equations.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Where 'Conic Section' Comes From

Every shape studied in this unit — parabolas, ellipses, and (in the next lesson) hyperbolas — gets its family name, conic section, from a genuinely geometric origin: each one is the curve traced by slicing a double cone with a flat plane at a different angle. Before reading on, think about a parabola not as 'the graph of a quadratic,' the way it's been treated since Algebra I, but as a set of points satisfying a distance condition — a definition that will also make sense of an ellipse.

Definition — Parabola (Focus-Directrix Definition)

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). For y² = 4px, the focus is at (p, 0) and the directrix is the line x = −p.

Worked Example — Finding a Parabola's Focus and Directrix

Find the focus and directrix of y² = 8x. Matching to y² = 4px: 4p = 8, so p = 2. The focus is at (2, 0), and the directrix is the line x = −2. The parabola opens to the right, toward the focus.

Definition — Ellipse

An ellipse is the set of all points where the sum of the distances to two fixed points (the foci) stays constant. Standard form: (x − h)²/a² + (y − k)²/b² = 1, centered at (h, k), with c² = a² − b² giving the distance from center to each focus (c).

The 'sum of distances stays constant' definition is exactly why an ellipse looks like a stretched circle — imagine a loop of string with both ends pinned at the two foci, traced taut with a pencil; every point the pencil reaches keeps the total string length (the sum of the two distances) fixed, tracing exactly an ellipse.

Worked Example — Identifying an Ellipse's Key Features

Identify the center, vertices, and foci of x²/9 + y²/4 = 1. Center: (0, 0). Since a² = 9 > b² = 4, the major axis is horizontal, with a = 3, giving vertices at (±3, 0). Co-vertices at (0, ±2), since b = 2. Foci: c² = a² − b² = 9 − 4 = 5, so c = √5, giving foci at (±√5, 0).

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

To plot an ellipse (or the upper/lower half of a parabola opening sideways) with a tool built for y = f(x) functions, solve the equation for y and graph the resulting positive and negative square-root branches together — the same split-branch technique used for circles.

Common Mistakes

Assuming the larger denominator always corresponds to the vertical axis, regardless of which variable it's under.
The major axis follows whichever variable (x or y) has the larger denominator — a larger denominator under x² means a horizontal major axis, not vertical.
Computing an ellipse's foci using c² = a² + b², the hyperbola relationship, instead of c² = a² − b².
An ellipse's foci use c² = a² − b² (foci sit inside the ellipse, closer to center than the vertices); a hyperbola instead uses c² = a² + b² — these are two different shapes with two different relationships.

Key Takeaways

A parabola is the set of points equidistant from a fixed focus and a fixed directrix line.
An ellipse is the set of points where the sum of distances to two foci stays constant.
An ellipse's foci are found using c² = a² − b², where a is the larger of the two axis measurements.

Summary

Parabolas and ellipses both trace back to genuine distance-based definitions, not just algebraic forms. The next lesson studies a third conic section — the hyperbola — whose equation looks deceptively similar to an ellipse's.

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