Lesson 72: Parabolas
A parabola is the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas appear in real-world applications such as satellite dishes, headlights, and projectile motion.
Equations of a Parabola
Standard forms depending on orientation:
- Vertical axis: (x − h)² = 4p(y − k), vertex at (h, k), focus at (h, k + p)
- Horizontal axis: (y − k)² = 4p(x − h), vertex at (h, k), focus at (h + p, k)
Here, p is the distance from the vertex to the focus (or vertex to directrix).
Examples
Example 1: Find the vertex, focus, and directrix of (x − 2)² = 8(y + 1)
Solution: Vertex = (2, −1), 4p = 8 → p = 2 Focus: (2, −1 + 2) = (2, 1), Directrix: y = −1 − 2 = −3
Example 2: Write the equation of a parabola with vertex at (0, 0), focus at (0, 3)
Solution: p = 3 → Equation: x² = 4(3)y → x² = 12y
Example 3: Determine the equation of a parabola opening left with vertex at (1, 2) and focus at (−2, 2)
Solution: Horizontal parabola, p = −3 → (y − 2)² = 4(−3)(x − 1) → (y − 2)² = −12(x − 1)
Practice Problems
- Find the vertex, focus, and directrix of (y + 3)² = 16(x − 1)
- Write the equation of a vertical parabola with vertex at (2, −1) and focus at (2, 2)
- A parabola has vertex at (0, 0) and directrix y = −4. Find its equation.
- Determine the focus and directrix for x² = 20y
- Write the equation of a horizontal parabola with vertex (−1, 3) and p = 5