Parabolas (Focus and Directrix)

In algebra you learned that a parabola is the graph of a quadratic function. In conic sections, we define a parabola geometrically: it is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). This definition reveals why satellite dishes, car headlights, and solar collectors are parabolic — the reflective property that makes these applications work comes directly from the focus-directrix relationship.

The Geometric Definition

A parabola is the set of all points $P$ in a plane such that:

$\text{distance}(P, \text{focus}) = \text{distance}(P, \text{directrix})$

The focus is a fixed point, and the directrix is a fixed line. The vertex is the point on the parabola closest to the directrix — it sits exactly halfway between the focus and the directrix. The distance from the vertex to the focus (or from the vertex to the directrix) is called $p$.

Standard Forms and the 4p Relationship

The distance $p$ controls the shape: larger $p$ means a wider parabola, smaller $p$ means a narrower one. There are four standard orientations.

Vertical Axis (Opens Up or Down)

$(x - h)^2 = 4p(y - k)$

• Vertex: $(h, k)$
• $p > 0$: opens upward — focus at $(h, k + p)$, directrix $y = k - p$
• $p < 0$: opens downward — focus at $(h, k + p)$, directrix $y = k - p$

Horizontal Axis (Opens Right or Left)

$(y - k)^2 = 4p(x - h)$

• Vertex: $(h, k)$
• $p > 0$: opens right — focus at $(h + p, k)$, directrix $x = h - p$
• $p < 0$: opens left — focus at $(h + p, k)$, directrix $x = h - p$

Key pattern: The squared variable tells you the axis of symmetry. $(x - h)^2$ means the axis is vertical; $(y - k)^2$ means the axis is horizontal.

Worked Examples

Example 1: Finding Focus and Directrix

Find the focus and directrix of $(x - 2)^2 = 12(y + 1)$.

Step 1: Identify the form. This is $(x - h)^2 = 4p(y - k)$, so the axis is vertical.

Step 2: Read the vertex: $(h, k) = (2, -1)$.

Step 3: Find $p$ from $4p = 12$: $p = 3$.

Since $p > 0$, the parabola opens upward.

• Focus: $(2, -1 + 3) = (2, 2)$
• Directrix: $y = -1 - 3 = -4$

Example 2: Writing the Equation from Focus and Directrix

Write the equation of the parabola with focus $(0, 5)$ and directrix $y = -5$.

Step 1: The vertex is halfway between focus and directrix:

$k = \frac{5 + (-5)}{2} = 0, \quad h = 0$

Vertex: $(0, 0)$.

Step 2: Find $p$. The focus is above the vertex, so the parabola opens upward:

$p = 5 - 0 = 5$

Step 3: Write the equation:

$x^2 = 4(5)y = 20y$

Example 3: Horizontal Parabola

Find the vertex, focus, and directrix of $(y + 3)^2 = -8(x - 1)$.

Step 1: This is $(y - k)^2 = 4p(x - h)$, so the axis is horizontal.

Step 2: Vertex: $(1, -3)$.

Step 3: $4p = -8$, so $p = -2$. The parabola opens left (since $p$ is negative).

• Focus: $(1 + (-2), -3) = (-1, -3)$
• Directrix: $x = 1 - (-2) = 3$

Example 4: Converting from General Form

Write $y = 2x^2 - 12x + 14$ in conic standard form and find the focus and directrix.

Step 1: Isolate the $y$ terms and complete the square on $x$:

$y - 14 = 2(x^2 - 6x)$

$y - 14 = 2(x^2 - 6x + 9 - 9) = 2(x - 3)^2 - 18$

$y + 4 = 2(x - 3)^2$

Step 2: Rewrite in conic form (solve for the squared term):

$(x - 3)^2 = \frac{1}{2}(y + 4)$

Step 3: $4p = \frac{1}{2}$, so $p = \frac{1}{8}$.

• Vertex: $(3, -4)$
• Focus: $\left(3, -4 + \frac{1}{8}\right) = \left(3, -\frac{31}{8}\right)$
• Directrix: $y = -4 - \frac{1}{8} = -\frac{33}{8}$

The small value of $p$ means this is a narrow parabola.

The Reflective Property

Any ray traveling parallel to the axis of symmetry reflects off the parabola and passes through the focus. This is why:

• Satellite dishes are parabolic — incoming parallel signals reflect to the receiver at the focus
• Car headlights place the bulb at the focus — light reflects outward in a parallel beam
• Solar collectors focus sunlight onto a pipe at the focus to heat fluid

The reflective property follows directly from the focus-directrix definition and the law of reflection (angle of incidence equals angle of reflection).

Real-World Application: Engineering a Reflector

An engineer is designing a parabolic solar reflector. The dish is 4 meters wide and 1 meter deep. Where should the collector pipe be placed?

Set up coordinates with the vertex at the origin and the parabola opening upward. The dish edge is at $(2, 1)$ (half-width of 2 m, depth of 1 m).

$(x)^2 = 4p(y)$

Substitute $(2, 1)$:

$4 = 4p(1) \Rightarrow p = 1$

The collector pipe should be placed 1 meter above the vertex (at the focus).

Common Mistakes

1. Confusing the squared variable. If $x$ is squared, the axis is vertical and the parabola opens up/down. If $y$ is squared, the axis is horizontal and it opens left/right.
2. Getting the sign of $p$ wrong. $p$ is positive when the parabola opens toward the positive direction (up or right), negative when it opens toward the negative direction (down or left).
3. Writing $2p$ instead of $4p$. The coefficient in the standard form is $4p$, not $2p$.
4. Placing the directrix on the wrong side. The directrix is always on the opposite side of the vertex from the focus.

Practice Problems

Key Takeaways

• A parabola is the set of all points equidistant from the focus and the directrix
• The distance from vertex to focus is $p$; the standard form coefficient is $4p$
• Vertical axis: $(x - h)^2 = 4p(y - k)$ — opens up ($p > 0$) or down ($p < 0$)
• Horizontal axis: $(y - k)^2 = 4p(x - h)$ — opens right ($p > 0$) or left ($p < 0$)
• The reflective property makes parabolas essential in engineering — satellite dishes, headlights, solar collectors
• Larger $|p|$ gives a wider parabola; smaller $|p|$ gives a narrower one

Return to College Algebra for more topics in this section.