Section 9.3 The Parabola.

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Presentation transcript:

Section 9.3 The Parabola

Objectives: Graph parabolas with vertices at the origin. Write equations of parabolas in standard form. Graph parabolas with vertices not at the origin. Solve applied problems involving parabolas.

Definition of a Parabola A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line.

Standard Forms of the Equations of a Parabola The standard form of the equation of a parabola with vertex at the origin is y2 = 4px or x2 = 4py

Example: Finding the Focus and Directrix of a Parabola Find the focus and directrix of the parabola given by y2 = 8x. Then graph the parabola.

The Latus Rectum and Graphing Parabolas The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. The length of the latus rectum for the graphs of y2 = 4px and x2 = 4py is

Example: Finding the Equation of a Parabola from Its Focus and Directrix Find the standard form of the equation of a parabola with focus (8, 0) and directrix x = –8.

Translations of Parabolas – Standard Forms of Equations of Parabolas with Vertex (h, k)

Example: Graphing a Parabola with Vertex at (h, k) Find the vertex, focus, and directrix of the parabola given by Then graph the parabola.

What is the equation of the parabola used to shape the mirror? Example: Application An engineer is designing a flashlight using a parabolic reflecting mirror and a light source. The casting has a diameter of 6 inches and a depth of 4 inches. What is the equation of the parabola used to shape the mirror? At what point should the light source be placed relative to the mirror’s vertex? 6 inches 4 inches

Example: Application (continued) We position the parabola with its vertex at the origin and opening upward. 6 inches 4 inches 4 inches 6 inches

Example: Application (continued) The focus is on the y-axis, located at (0, p). The form of the equation is The point (3, 4) lies on the parabola. To find p, we let x = 3 and y = 4.

Example: Application (continued) We substitute in to obtain the standard form of the equation of the parabola. The equation of the parabola used to shape the mirror is The light source should be placed at or inches above the vertex.