1. Section 10.1
The Parabola
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2. Objectives
Given an equation of a parabola, complete the square, if necessary, and then find the vertex, the focus, and the directrix and graph the parabola.

3. Conic Sections
A conic section is formed when a right circular cone with two parts, called nappes is intersected by a plane. One of four types of curves can be formed: a parabola, a circle, an ellipse, or a hyperbola.

4. Parabolas
A parabola is the set of all points in a plane equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus).
The line that is perpendicular to the directrix and contains the focus is the axis of symmetry. The vertex is the midpoint of the segment between the focus and the directrix.

5. Standard Equation of a Parabola with Vertex at the Origin
The standard equation of a parabola with vertex (0, 0) and directrix y = p is x2 = 4py.
The focus is (0, p) and the y-axis is the axis of symmetry.

6. Standard Equation of a Parabola with Vertex at the Origin
The standard equation of a parabola with vertex (0, 0) and directrix x = p is y2 = 4px.
The focus is (p, 0) and the x-axis is the axis of symmetry.

7. Example
Find the focus, the vertex and the directrix of the parabola Then graph the parabola.
We write in the form x2 = 4py:
Thus, p =  3. Focus is (0, p), or (0,  3). Directrix is y = p = 3.

8. Example
Find an equation of the parabola with vertex (0, 0) and focus (5, 0). Then graph the parabola.
The focus is on the x-axis so the line of symmetry is the x-axis. Thus the equation is of the type y2 = 4px.
Since the focus (5, 0) is 5 units to the right of the vertex, p = 5 and the equation is y2 = 4(5)x, or y2 = 20x.

9. Standard Equation of a Parabola with Vertex (h, k) and Vertical Axis of Symmetry
The standard equation of a parabola with vertex (h, k) and vertical axis of symmetry is
where the vertex is (h, k), the focus is (h, k + p), and the directrix is y = k – p.

10. Standard Equation of a Parabola with Vertex (h, k) and Horizontal Axis of Symmetry
The standard equation of a parabola with vertex (h, k) and horizontal axis of symmetry is
where the vertex is (h, k), the focus is (h + p, k), and the directrix is x = h – p.

11. Example
For the parabola x2 + 6x + 4y + 5 = 0, find the vertex, the focus, and the directrix. Then draw the graph.
We first complete the square:

12. Example (continued)
We see that h = –3, k = 1 and p = –1, so we have the following:
Vertex (h, k): (–3, 1);
Focus (h, k + p): (–3, 1 + (–1)) or (3, 0);
Directrix y = k  p: y = 1  (–1), or y = 2.

13. Example
For the parabola y2 – 2y – 8x – 31 = 0, find the vertex, the focus, and the directrix. Then draw the graph.
We first complete the square:

14. Example (continued)
We see that h = –4, k = 1 and p = 2, so we have the following:
Vertex (h, k): (–4, 1);
Focus (h, k + p): (–4 + 2, 1) or (–2, 1);
Directrix x = h  p: y = –4  2, or x = –6.

15. Applications
Cross sections of car headlights, flashlights, search lights: the bulb is at the focus, light rays from that point are reflected outward parallel to axis of symmetry.

16. Applications (continued)
Cross sections of satellite dishes, field microphones: incoming radio waves or sound waves parallel to the axis of symmetry are reflected into the focus.

17. Applications (continued)
Cables hung between structures in suspension bridges, such as the Golden Gate Bridge. When a cable supports only its own weight, it forms a curve called a catenary.