2-3: Focus of a Parabola Explore the focus and the directrix of a parabola. Write equations of parabolas.

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2-3: Focus of a Parabola Explore the focus and the directrix of a parabola. Write equations of parabolas.

Focus and Directrix Parabola: the set of all points (x, y) in a plane that are equidistant from a fixed point, called the focus and a fixed line, called the directrix. Focus: the fixed point that is in the interior of the parabola and lies on the axis of symmetry. Directrix: A fixed line that lies |p| units from the vertex outside of the parabola. |p| is also the distance from the focus to the vertex of the parabola.

Example 1 Use the Distance Formula to write an equation of the parabola with focus F(0, 4) and directrix y = −4. F(0,4) P(x, y) y = -4 D(x,-4)

Equation of Parabola that opens up or down with vertex (0,0), focus at (0,p) and the directrix y = -p: (x, y) F(0,p) (x, -p) y = - p

Standard Equation of Parabola with vertex at the origin Vertical axis of symmetry: x = 0 Parabola opens up, p>0 Parabola opens down, p<0 Horizontal axis of symmetry: y = 0 Parabola opens left, p<0 Parabola opens right, p>0

Example 2 Graph the equation of the given parabola and identify the focus, directrix, and axis of symmetry. Step 1: Rewrite the equation in Standard Form: Step 2: Identify the focus, directrix, and axis of symmetry Focus: Directrix: Axis of Symmetry:

Example 3 Write the equation of the parabola shown. Vertex at (0,0), parabola faces down, So standard equation of Parabola: Directrix Vertex The directrix is y = 1.5, so p = -1.5

Standard Equations of a Parabola with Vertex (h, k) Vertex is at (h,k) Focus will be p units above : Vertex is at (h,k) Focus will be p units right: Directrix will be p units down Directrix will be p units left Axis of Symmetry: Axis of Symmetry:

Example 4 Write the equation of the graph. Parabola faces right

Example 5 An electricity-generating dish uses a parabolic reflector to concentrate sunlight onto a high frequency engine at the focus. Write the equation that represents the cross section of the dish with the vertex being (0,0). What is the depth of the dish? The depth is the height of the dish at the edge.