AP PRECALCULUS — UNIT 4A · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.6A — Conic Sections — Parabolas
Notes — Focus, Directrix, and Standard Form
💡 Learning Objectives (4.6.A Part 1)
By the end of this lesson you will be able to:
- Define a parabola in terms of focus and directrix
- Identify the vertex, focus, directrix, axis of symmetry of a parabola
- Write the standard form equation of a parabola from given features
- Sketch a parabola given its equation in standard form
1. Conic Sections at a Glance
Conic sections are curves formed by slicing a cone with a plane. The four basic types are:
- Circle (slice perpendicular to the axis)
- Ellipse (slice at a moderate angle)
- Parabola (slice parallel to the slant of the cone)
- Hyperbola (steep slice, cutting both nappes)
Algebraically, all four can be written in the form Ax² + Bxy + Cy² + Dx + Ey + F = 0. We'll handle each in standard form, where the curves are positioned simply.
2. Definition of a Parabola
A parabola is the set of all points equidistant from a fixed point (the FOCUS) and a fixed line (the DIRECTRIX).
The line through the focus perpendicular to the directrix is the AXIS OF SYMMETRY. The point on the axis halfway between focus and directrix is the VERTEX.
3. Standard Forms
💡 Standard Forms of a Parabola
- Vertical axis (opens up or down): (x − h)² = 4p(y − k). Vertex (h, k); focus (h, k + p); directrix y = k − p. Opens UP if p > 0, DOWN if p < 0.
- Horizontal axis (opens right or left): (y − k)² = 4p(x − h). Vertex (h, k); focus (h + p, k); directrix x = h − p. Opens RIGHT if p > 0, LEFT if p < 0.
The parameter p is the directed distance from the vertex to the focus. |p| is also the distance from the vertex to the directrix.
4. Worked Examples
📘 Example — Identify features of (x − 3)² = 8(y − 1)
- Vertex: (3, 1)
- 4p = 8, so p = 2
- Opens UP (p > 0)
- Focus: (3, 1 + 2) = (3, 3)
- Directrix: y = 1 − 2 = −1
- Axis of symmetry: x = 3
📘 Example — Write the equation
Parabola with vertex (−1, 4) and focus (−1, 6).
- Same x-coordinate ⇒ vertical axis
- p = 6 − 4 = 2 (focus is above vertex)
- Equation: (x − (−1))² = 4(2)(y − 4), i.e. (x + 1)² = 8(y − 4)
5. The Reflective Property
A parabola has a beautiful reflection property: any incoming ray parallel to the axis of symmetry reflects off the curve and passes through the FOCUS. This is why satellite dishes, headlights, and telescope mirrors are parabolic — they focus or distribute energy through the focus.
6. Sketching
- Plot the vertex
- Plot the focus (inside the curve)
- Draw the directrix (a line outside the curve, opposite the focus)
- Plot two more points using the latus rectum (the chord through the focus perpendicular to the axis): its length is |4p|, so two points at distance |2p| from the focus on either side
- Connect with a smooth U-shape opening toward the focus
7. Converting from General Form
If a parabola is given as ax² + bx + cy + d = 0 (or similar), complete the square to put it into standard form.
📘 Example — Convert x² − 4x + 4y + 8 = 0
- Group: x² − 4x = −4y − 8
- Complete the square: (x − 2)² − 4 = −4y − 8
- Rearrange: (x − 2)² = −4y − 4 = −4(y + 1)
- Standard form: (x − 2)² = −4(y + 1). Vertex (2, −1); 4p = −4 so p = −1; opens DOWN; focus (2, −2); directrix y = 0
8. Summary
- Parabola: equidistant set from a focus and a directrix
- Vertical axis: (x − h)² = 4p(y − k); horizontal axis: (y − k)² = 4p(x − h)
- p = signed distance vertex-to-focus; |p| = distance vertex-to-directrix
- Sign of p tells the direction of opening
- Reflective property: parallel rays reflect through the focus