Equation of a Parabola
Aligned to G-GPE.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
A parabola is the set of points equidistant from a fixed focus and a fixed line (the directrix). Setting the distance to the focus equal to the distance to the directrix and simplifying gives the equation. With focus (0, p) and directrix y = -p, the parabola is y = x^2/(4p).
Worked example
Find the equation with focus (0, 2) and directrix y = -2. A point (x, y) is equidistant from both. Distance to focus: sqrt(x^2 + (y - 2)^2). Distance to directrix: y + 2. Set equal and square: x^2 + (y - 2)^2 = (y + 2)^2. Expand: x^2 + y^2 - 4y + 4 = y^2 + 4y + 4. Simplify: x^2 = 8y, so y = x^2/8.
Practice questions
- A parabola has focus (0, 3) and directrix y = -3. Set up the equal-distance equation.
- Derive the equation for focus (0, 1) and directrix y = -1.
- For y = x^2/12, identify the value of p and state the focus and directrix.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Ecuación de una parábola
Una parábola es el conjunto de puntos equidistantes de un foco fijo y una recta fija (la directriz). Igualar la distancia al foco con la distancia a la directriz y simplificar da la ecuación. Con foco (0, p) y directriz y = -p, la parábola es y = x^2/(4p).
Ejemplo: Halla la ecuación con foco (0, 2) y directriz y = -2. Un punto (x, y) equidista de ambos. Distancia al foco: raíz(x^2 + (y - 2)^2). Distancia a la directriz: y + 2. Iguala y eleva al cuadrado: x^2 + (y - 2)^2 = (y + 2)^2. Desarrolla: x^2 + y^2 - 4y + 4 = y^2 + 4y + 4. Simplifica: x^2 = 8y, así que y = x^2/8.
More Expressing Geometric Properties with Equations lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required