Parabolas
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 7.5: “Conic Sections” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/7-5-conic-sections |
| Textbook used in class | Stewart, Calculus, Section 10.5: “Conic Sections” |
Opening Scenario
Satellite dishes, car headlights, and parabolic reflectors all exploit the same geometric property: any ray coming in parallel to the axis of a parabola reflects through the focus. This reflective property follows directly from the definition of the parabola as a locus of equidistant points.
Quick Reference
Definition: A parabola is the set of all points equidistant from a fixed point (the focus $F$) and a fixed line (the directrix $\ell$).
Standard forms with vertex at the origin:
| Equation | Axis | Focus | Directrix | Opens |
|---|---|---|---|---|
| $x^2 = 4py$ | $y$-axis | $(0,p)$ | $y = -p$ | Up if $p>0$, down if $p<0$ |
| $y^2 = 4px$ | $x$-axis | $(p,0)$ | $x = -p$ | Right if $p>0$, left if $p<0$ |
Shifted parabola with vertex $(h,k)$: replace $x$ with $x-h$ and $y$ with $y-k$.
Key Concepts
1. Deriving the Standard Form
Place the focus at $(0,p)$ and directrix at $y = -p$. A point $(x,y)$ is on the parabola iff: $$\sqrt{x^2+(y-p)^2} = |y+p|.$$
Square: $x^2+(y-p)^2 = (y+p)^2$.
$x^2 + y^2 - 2py + p^2 = y^2 + 2py + p^2$.
$x^2 = 4py$.
2. Reading the Parameters
Given $x^2 = 4py$:
- If $4p > 0$: opens upward; if $4p < 0$: opens downward.
- Vertex at $(0,0)$; focus at $(0,p)$; directrix $y = -p$.
- The absolute value $|p|$ is the distance from vertex to focus (and from vertex to directrix).
Given $y^2 = 4px$:
- If $4p > 0$: opens right; if $4p < 0$: opens left.
- Vertex at $(0,0)$; focus at $(p,0)$; directrix $x = -p$.
3. The Reflective Property
If a ray travels parallel to the axis of the parabola and hits the parabola, the reflected ray passes through the focus. Equivalently, rays emanating from the focus reflect off the parabola parallel to the axis. This is why parabolic mirrors concentrate light at the focus (telescope) or direct light into a beam (flashlight).
Worked Example
Find the focus, directrix, and axis of $y^2 = -12x$. Sketch the parabola.
The equation has the form $y^2 = 4px$ with $4p = -12$, so $p = -3$.
- Opens: to the left (since $p < 0$).
- Vertex: $(0,0)$.
- Axis: $y = 0$ (the $x$-axis).
- Focus: $(p,0) = (-3,0)$.
- Directrix: $x = -p = 3$.
Verification: The point $(-3, 6)$ should be on the parabola since $6^2 = 36$ and $(-12)(-3) = 36$. Distance to focus: $\sqrt{(-3-(-3))^2+(6-0)^2} = 6$. Distance to directrix $x=3$: $|{-3-3}| = 6$. Equal. $\checkmark$
Boxed answers: Opens left; vertex $(0,0)$; focus $(-3,0)$; directrix $x = 3$.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Confusing $4p$ with $p$ | Reporting focus at $(12,0)$ for $y^2 = 12x$ | $4p = 12 \Rightarrow p = 3$; focus is at $(3,0)$, not $(12,0)$ |
| Getting the sign of the directrix wrong | Writing directrix as $x = p$ instead of $x = -p$ | Focus and directrix are on opposite sides of the vertex; focus at $(p,0)$ means directrix at $x = -p$ |
| Confusing horizontal and vertical parabolas | Treating $y^2 = 4px$ as opening up/down | $y^2 = 4px$ opens left/right; $x^2 = 4py$ opens up/down |
Common Misconceptions
the focus of the parabola $x^2 = 4py$ is at $(0, 4p)$ rather than $(0, p)$.
This is the input-output-confusion error in reading the standard form. The equation $x^2 = 4py$ uses $4p$ as the coefficient of $y$, so $p$ must be extracted by dividing that coefficient by $4$. Reporting the focus at $(0, 4p)$ treats the entire coefficient as $p$, quadrupling the actual focal distance. For $x^2 = 8y$, the coefficient is $8 = 4p$ so $p = 2$ and the focus is at $(0, 2)$, not $(0, 8)$.
$y^2 = 4px$ opens upward or downward.
This is the iconic-graph error. The equation $y^2 = 4px$ has $y$ squared, so it is symmetric about the $x$-axis and opens to the right ($p > 0$) or left ($p < 0$). Only $x^2 = 4py$ opens upward or downward. Students who pattern-match on the presence of a squared variable without noting which variable is squared routinely misidentify the axis of the parabola.
Leveled Practice
Level 1 -- Identify Features
Problem 1. Find the vertex, focus, and directrix of $x^2 = 8y$.
Show answer
$4p = 8 \Rightarrow p = 2$.
Vertex: $(0,0)$. Focus: $(0,2)$. Directrix: $y = -2$. Opens upward.
Level 2 -- Find the Equation
Problem 2. Find the equation of the parabola with vertex at the origin and focus at $(0,-5)$.
Show answer
Focus at $(0,-5)$ means $p = -5$ and the axis is vertical.
Equation: $x^2 = 4py = 4(-5)y = -20y$.
$x^2 = -20y$.
Mastery Checklist
Mental Model
A parabola is the “fairness” curve: every point on it is equally far from the focus and the directrix. The vertex is at the midpoint between focus and directrix. The parameter $p$ is that half-distance. Increasing $|p|$ makes the parabola wider and more open (focus farther from vertex); decreasing $|p|$ makes it narrower (focus close to vertex, steep walls).
Connections
Looking back
- Completing the square: Used to put a non-standard parabola $x^2 + bx + cy = d$ in vertex form.
Looking ahead
- Ellipses and hyperbolas (Section 10.5): The same focus-centric definitions generalize.
- Conics in polar form (Section 10.6): Parabolas have eccentricity $e = 1$ in the unified polar formula.