The Parabola

You already know the parabola as the graph of $y = ax^2 + bx + c$. What you may not know is that the same shape has a purely geometric definition that does not mention algebra: a parabola is the set of all points equidistant from a fixed point and a fixed line. The fixed point is the focus; the fixed line is the directrix. That definition explains why parabolic mirrors focus incoming sunlight onto a single hot spot, why a flashlight reflector throws a parallel beam, and why a projectile in flight traces a parabola against gravity. Understanding the focus-directrix definition unlocks all of those applications.

Two forms of the equation

The standard quadratic form is $y = ax^2 + bx + c$ with $a \neq 0$. The parabola opens upward when $a > 0$, downward when $a < 0$. Its vertex sits at $x = -b/(2a)$; substitute back to get the $y$-coordinate.

The vertex form $y = a(x - h)^2 + k$ makes the vertex $(h, k)$ explicit. The two forms are linked by expanding the square: $b = -2ah$ and $c = ah^2 + k$. Completing the square moves you from standard to vertex form.

Focus and directrix

For the vertex-form parabola $y = a(x - h)^2 + k$, define $p = 1/(4a)$. Then:

• The focus is at $(h,\; k + p)$.
• The directrix is the horizontal line $y = k - p$.
• $|p|$ is the vertex-to-focus distance.

An alternative canonical form when the vertex is at the origin and the parabola opens upward is $x^2 = 4py$, with focus $(0, p)$ and directrix $y = -p$. The formula falls straight out of the focus-directrix definition: setting "distance from $(x, y)$ to $(0, p)$" equal to "distance from $(x, y)$ to the line $y = -p$" and squaring both sides yields exactly $x^2 = 4py$.

• Physics: A thrown ball's trajectory under gravity (ignoring air drag) is a parabola; its vertex is the apex of the throw, and its focus appears in the formula for the time-of-flight of light reflected off a parabolic mirror.
• Engineering: Satellite dishes and car headlights are parabolic reflectors: the directrix and focus are the geometric guarantee that every parallel ray hitting the dish converges exactly at the focus.
• Architecture: Suspension-bridge cables hang in a catenary (not a parabola), but the load-bearing curve when the deck is uniformly loaded IS a parabola, the Golden Gate's deck-load curve is a textbook example.

(Select "parabola" mode and slide $p$ up and down. The focus and directrix move together; the vertex sits at the midpoint. Try making the parabola very flat and then very narrow.)

Axis of symmetry

Every parabola is symmetric about a line through its vertex perpendicular to the directrix. In vertex form that line is $x = h$; in standard form it is $x = -b/(2a)$. The vertex is the unique point on the parabola where the curve touches its axis of symmetry, the minimum (if it opens upward) or maximum (if downward).

Why physics loves parabolas

A projectile under constant gravity has horizontal position linear in time and vertical position quadratic in time: eliminate time, and the trajectory is a parabola in space. A parabolic mirror reflects rays parallel to its axis through the focus, the optical property exploited by satellite dishes, headlights, and radio telescopes. Both applications come from the same focus-directrix geometry.

Try it

• Find the vertex, focus, and directrix of $y = 2x^2 - 8x + 5$. (Complete the square first.)
• A parabola has vertex at the origin and focus at $(0, 3)$. Find its equation.
• Does $y = -x^2 + 6x - 1$ open up or down? Locate its vertex.

Pause: as $a$ gets larger and larger, what happens to $p = 1/(4a)$? The focus moves closer to the vertex, and the parabola gets narrower. A wide, flat parabola has its focus far away.

A trap to watch for

Beginners often write the focus at $(h, p)$ instead of $(h, k + p)$. The error: forgetting to add $k$, the vertex's $y$-coordinate. The focus is always offset from the vertex by $p$ along the axis of symmetry, it is not at $(h, p)$ unless the vertex happens to be at the origin. Double-check by reading the vertex form $y = a(x - h)^2 + k$: the $h$ and $k$ tell you where the vertex sits, and the focus lives directly above (or below) it by $p$ units.

A second trap: confusing the sign of $p$. When the parabola opens downward ($a < 0$), $p$ is negative, the focus is below the vertex, and the directrix is above. Always check the sign of $a$ before placing the focus.

What you now know

You can convert between standard and vertex forms by completing the square, identify the vertex, axis of symmetry, focus, and directrix, sketch the parabola opening up or down, and derive the canonical equation from the focus-directrix definition. The next section introduces the ellipse, the conic where every point is at a constant sum of distances from two foci.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 12, §2, parabolas defined geometrically and algebraically.
• Apostol, T. M. (1969). Calculus, Volume 2. Wiley. Chapter 13: conic sections including the focus-directrix definition.
• Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. Chapter 6: conics from the classical Greek perspective.