Conic Sections Parabola Conic Sections Parabola The parabola

Conic Sections Parabola

Conic Sections - Parabola The parabola has the characteristic shape shown above. A parabola is defined to be the “set of points the same distance from a point and a line”.

Conic Sections - Parabola Focus Directrix The line is called the directrix and the point is called the focus.

Conic Sections - Parabola Focus Vertex Axis of Symmetry Directrix The line perpendicular to the directrix passing through the focus is the axis of symmetry. The vertex is the point of intersection of the axis of symmetry with the parabola.

Conic Sections - Parabola Focus d 1 d 2 Directrix The definition of the parabola is the set of points the same distance from the focus and directrix. Therefore, d 1 = d 2 for any point (x, y) on the parabola.

Finding the Focus and Directrix Parabola

Conic Sections - Parabola Therefore, the distance p from the vertex to the focus and the vertex to the directrix is given by the formula

Conic Sections - Parabola Using transformations, we can shift the parabola y=ax 2 horizontally and vertically. If the parabola is shifted h units right and k units up, the equation would be The vertex is shifted from (0, 0) to (h, k). Recall that when “a” is positive, the graph opens up. When “a” is negative, the graph reflects about the x-axis and opens down.

Example 1 Graph a parabola. Find the vertex, focus and directrix.

Parabola – Example 1 Make a table of values. Graph the function. Find the vertex, focus, and directrix.

Parabola – Example 1 The vertex is (-2, -3). Since the parabola opens up and the axis of symmetry passes through the vertex, the axis of symmetry is x = -2.

Parabola – Example 1 Make a table of values. x y -2 -3 -1 0 1 Plot the points on the graph! Use the line of symmetry to plot the other side of the graph. 2 3 4 -1