Section 9.3 The Parabola

Finally, something familiar! The parabola is oft discussed in MTH 112, as it is the graph of a quadratic function: Does look familiar? Our discussion of the parabola will be consistent with our discussion of the other conic sections.

The Parabola A parabola is the set of all points in the plane that are the same distance from a given point (focus) as they are from a given line (directrix). It is important to note that the focus is not a point on the directrix. While the directrix can be any line, we only consider horizontal and vertical ones.

Parabolic Parts The axis of symmetry of a parabola is an imaginary line perpendicular to the directrix that passes through the focus. The axis of symmetry intersects the parabola at a point called the vertex. Let p be the distance from the vertex to the focus. It follows that p is also the distance from the vertex to the directrix.

More Pictures

Equations The standard form of the equation of a parabola with horizontal directrix is When p is positive, the parabola opens upward. When p is negative, the parabola opens downward.

Equations (cont.) The standard form of the equation of a parabola with vertical directrix is When p is positive, the parabola opens to the right. When p is negative, the parabola opens to the left.

Giggle, giggle The latus rectum is a line segment that: 1.passes through the focus; 2.Is parallel to the directrix; 3.Has its endpoints on the parabola. The length of the latus rectum is 4p.

Pictures

Finally… Draw the picture. And, when all else fails…. Draw the picture.

Examples Find the vertex, focus and directrix, and sketch the graph. 1.x 2 = 24y 2.y 2 = 40x 3.(x + 2) 2 = -4(y – 1) Find the standard form of the equation the parabola so described: 1.Focus is (12, 0); directrix is x = Vertex is (3, -1); focus is (3, -2)

More Examples Convert to standard form by completing the square on x: x 2 + 6x – 12y – 15 = 0 Convert to standard form by completing the square on y: y 2 – 12y + 16x + 36 = 0