Introduction to Conics: Parabolas

Transcription

1 Introduction to Conics: Parabolas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011

2 Objectives In this lesson we will learn to: recognize a conic as the intersection of a plane and a double-napped cone, write the equations of parabolas in standard form and graph parabolas, use the reflective property of parabolas to solve real-world problems.

3 Double-Napped Cone

4 Circle

5 Ellipse

6 Parabola

7 Point

8 Hyperbola

9 Line

10 Two Intersecting Lines

11 Geometric Property of a Parabola Definition A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line (called the directrix) and a fixed point (called the focus) not on the line. Remarks: 1 The midpoint between the focus and the directrix is called the vertex. 2 The line passing through the focus and the vertex is called the axis of the parabola.

12 Illustration Axis d 2 Focus Vertex d 1 d 1 d2 Directrix

13 Standard Form Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at (h, k) is as follows: (x h) 2 = 4p(y k), p 0 Vertical axis, directrix: y = k p (y k) 2 = 4p(x h), p 0 Horizontal axis, directrix: x = h p The focus lies on the axis p units (directed distance) from the vertex. If the vertex is at the origin (0, 0), the equation takes on one of the following forms. x 2 = 4py Vertical axis y 2 = 4px Horizontal axis

14 Illustration (x h) 2 = 4p(y k) : p > 0 Axis: x h Focus: h,k p Vertex: h,k Directrix: y k p

15 Example Find the standard form of the equation of a parabola whose vertex is at the origin and whose focus is at ( 3/2, 0).

16 Example Find the standard form of the equation of a parabola whose vertex is at the origin and whose focus is at ( 3/2, 0). The axis of the parabola is horizontal, thus we seek an equation of the form: (y k) 2 = 4p(x h). The vertex is at (0, 0) = (h, k) and the focus is at ( 3/2, 0) = (h + p, k) which implies p = 3/2. Therefore y 2 = 6x.

17 Example Find the vertex, focus, and directrix of the parabola: y 2 4y 4x = 0

18 Example Find the vertex, focus, and directrix of the parabola: y 2 4y 4x = 0 We must write the equation of the parabola in standard form. y 2 4y = 4x y 2 4y + 4 = 4x + 4 (y 2) 2 = 4(x + 1) Note that h = 1, k = 2, and p = 1. Vertex: (h, k) = ( 1, 2) Focus: (h + p, k) = (0, 2) Directrix: x = h p = 2

19 Example Find the standard form of the equation of the parabola with vertex at (1, 2) and directrix y = 1. Also write the quadratic equation for the parabola.

20 Example Find the standard form of the equation of the parabola with vertex at (1, 2) and directrix y = 1. Also write the quadratic equation for the parabola. Let (h, k) = (1, 2) and y = k p = 1 which implies p = 3. (x h) 2 = 4p(y k) (x 1) 2 = 4(3)(y 2) (x 1) 2 = 12(y 2)

21 Example Find the standard form of the equation of the parabola with vertex at (1, 2) and directrix y = 1. Also write the quadratic equation for the parabola. Let (h, k) = (1, 2) and y = k p = 1 which implies p = 3. (x h) 2 = 4p(y k) (x 1) 2 = 4(3)(y 2) (x 1) 2 = 12(y 2) x 2 2x + 1 = 12y 24 12y = x 2 2x + 25 y = 1 12 x x

22 Application: Reflective Property Rays parallel to the axis of the parabola will be reflected off the surface of the parabola toward the focus. Rays starting at the focus in any direction will be reflected off the parabola and will exit parallel to the axis of the parabola.

23 Homework Read Section 6.2. Exercises: 1, 5, 9, 13,..., 53, 57