Algebra 2 · Unit 8 · Topic 8.3
Parabolas as Distance Loci
A parabola consists of points equally distant from a focus and a directrix.
Definition and explanation
A parabola consists of points equally distant from a focus and a directrix. In (x − h)² = 4p(y − k) with p ≠ 0, the vertex is (h, k), the focus is (h, k + p), and the directrix is y = k − p.
Worked example
For x² = 12y, 4p = 12 gives p = 3. The vertex is (0, 0), the focus is (0, 3), and the directrix is y = −3. The point (6, 3) lies on the curve and is 6 units from both the focus and directrix.
Common mistake
The coefficient 12 is 4p, not the focal distance p.
Key takeaway
A parabola consists of points equally distant from a focus and a directrix.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.