3 Conic Sections - Parabola The intersection of a plane with one nappe of the cone is a parabola.
4 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”.
5 Conic Sections - Parabola FocusDirectrixThe line is called the directrix and the point is called the focus.
6 Conic Sections - Parabola Axis of SymmetryFocusVertexDirectrixThe 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.
7 Conic Sections - Parabola Focusd1Directrixd2The definition of the parabola is the set of points the same distance from the focus and directrix. Therefore, d1 = d2 for any point (x, y) on the parabola.
8 Conic Sections – Parabola (Unit 3-7) Using transformations, we can shift the parabola y=ax2 horizontally and vertically. If the parabola is shifted h units right and k units up, the equation would beThe 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.
9 Parabola – Example 1Identify the vertex, axis of symmetry, and direction of opening of a parabola. Graph the equation.
10 Parabola – Example 1The 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.
12 Paraboloid Revolution A paraboloid revolution results from rotating a parabola around its axis of symmetry as shown at the right.GNU Free Documentation License
13 Paraboloid Revolution They are commonly used today in satellite technology as well as lighting in motor vehicle headlights and flashlights.
14 Paraboloid Revolution The focus becomes an important point. As waves approach a properly positioned parabolic reflector, they reflect back toward the focus. Since the distance traveled by all of the waves is the same, the wave is concentrated at the focus where the receiver is positioned.
15 Example Problems 2x2 + 8x – 3 + y = 0 Find the vertex, axis of symmetry and direction of opening of the parabola. Then graph the equation and state the domain and range.