Advanced Functions

Conic Sections

Slice a cone with a plane at different angles and you get four distinct curves: circles, ellipses, parabolas, and hyperbolas. These conic sections were studied by the ancient Greeks, and they turn out to describe everything from planetary orbits to satellite dishes to the arches of bridges.

The Circle: Where It All Starts

A circle is the set of all points at a fixed distance (the radius) from a center point. The equation for a circle centered at the origin with radius r is:

x^2 + y^2 = r^2

We can plot this by graphing the upper and lower halves as separate functions:

Radius r2

0.54

x^2 + y^2 = 2^2

The circle is perfectly symmetric in every direction. Its eccentricity is 0 — it is the most “round” a conic section can be.

Connection

A circle is actually a special case of an ellipse where both axes are equal. As you stretch one axis, the circle becomes an ellipse. The eccentricity measures how much the shape has deviated from a perfect circle.

The Ellipse: A Stretched Circle

An ellipse centered at the origin has the equation:

x^2/a^2 + y^2/b^2 = 1

where a is the semi-major axis (half the width) and b is the semi-minor axis (half the height). When a = b, the ellipse is a circle.

a (semi-major)3

0.55

b (semi-minor)2

0.55

\frac{x^2}{ 3^2} + \frac{y^2}{ 2^2} = 1

The purple curve is the ellipse. The red horizontal lines mark the height of the semi-minor axis at plus and minus b.

Try This

Set a = b (for example, both equal to 3). The ellipse becomes a perfect circle. Now slowly increase a while keeping b fixed. Watch the ellipse stretch horizontally. The more different a and b are, the more elongated the ellipse becomes.

Eccentricity: Measuring Elongation

The eccentricity of an ellipse (assuming a >= b) is:

e = sqrt(1 - b^2/a^2)

Eccentricity ranges from 0 (a perfect circle) to values approaching 1 (an extremely elongated ellipse).

a (semi-major)3

b (semi-minor)2

0.35

e = \sqrt{1 - \frac{ 2^2}{ 3^2}}

Connection

Planetary orbits are ellipses with the Sun at one focus. Earth’s orbit has eccentricity about 0.017 — nearly a perfect circle. Pluto’s orbit has eccentricity 0.25, making it noticeably elongated. Comets can have eccentricities very close to 1, producing long narrow orbits.

The Hyperbola: Two Branches

A hyperbola looks like two mirror-image curves that open outward. The equation is:

x^2/a^2 - y^2/b^2 = 1

The minus sign (instead of plus) is what makes a hyperbola instead of an ellipse.

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b1.5

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\frac{x^2}{ 2^2} - \frac{y^2}{ 1.5^2} = 1

The yellow lines are the asymptotes — the hyperbola approaches these lines but never touches them. The asymptotes have slopes b/a and -b/a, forming an X shape that guides the curve’s direction.

The hyperbola has two branches, one opening right and one opening left.
The vertices (closest points of the two branches) are at (a, 0) and (-a, 0).
The gap between the branches is 2a wide.

Try This

Set a = b. The asymptotes become y = x and y = -x, forming a perfect 90-degree angle. This special case is called a rectangular hyperbola. The function y = 1/x from the rational functions lesson is actually a rectangular hyperbola rotated 45 degrees.

Parabolas Revisited

A parabola is also a conic section — it is the case where the slicing plane is parallel to the side of the cone. You already know parabolas from quadratic functions, but in conic section form, we write:

y = (1/4p) x^2 where p is the distance from vertex to focus.

p (focus distance)1

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y = \frac0{ 4 \cdot 1} x^2 \quad \text{focus at } (0, 1)

A larger p makes the parabola wider (the focus is farther from the vertex).
A smaller p makes it narrower and steeper.
The directrix (yellow line) sits at y = -p, below the vertex.

Connection

Satellite dishes and car headlights use parabolic reflectors. Any signal arriving parallel to the axis reflects off the parabolic surface and converges at the focus. That is why the receiver is mounted at the focal point of the dish.

The Family Portrait

All four conics come from the same formula. The general second-degree equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 produces:

Conic	Condition	Eccentricity
Circle	A = C, B = 0	e = 0
Ellipse	A and C same sign, A is not C	0 < e < 1
Parabola	A = 0 or C = 0 (not both)	e = 1
Hyperbola	A and C opposite signs	e > 1

Challenge

Challenge: Classify each equation as a circle, ellipse, parabola, or hyperbola:

x^2 + y^2 = 25
4x^2 + 9y^2 = 36
x^2 - y^2 = 16
y = 3x^2 + 2

Then rewrite each in standard form and identify the key features (center, vertices, asymptotes, etc.).

Take the Quiz