Today in Precalculus

1. Today in Precalculus • Notes: Conic Sections - Parabolas • Homework • Go over test

2. Conic Sections Conic sections are formed by the intersection of a plane and a cone. hyperbola circle ellipse parabola

3. Degenerate Conic Sections Atypical conics The conic sections can be defined algebraically in the Cartesian plane as the graphs of second-degree equations in two variables, that is, equations of the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, and C are not all zero intersecting lines point line

4. Parabolas Definition: A parabola is the set of all points in a plane equidistant from the directrix and the focus in the plane. The line passing through the focus and perpendicular to the directrix is the axis of the parabola and is the line of symmetry for the parabola. The vertex is midway between the focus and the directrix and is the point of the parabola closest to both. axis focus vertex directrix

5. P(x,y) F(0,p) By definition, the distance between F and P has to equal the distance between P and D. x2 +y2 – 2py + p2 = y2 + 2py +p2 x2 = 4py The standard form of the equation of a parabola that opens upward or downward D(x,-p)

6. If p>0, the parabola opens upward, if p<0 it opens downward. Parabolas that open to the left or right are inverse relations of upward or downward opening parabolas. So equations of parabolas with vertex (0,0) that open to the right or to the left have the standard form y2 = 4px If p>0, the parabola opens to the right and if p<0, the parabola opens to the left.

7. p is the focal length of the parabola – the directed distance from the vertex to the focus of the parabola. A line segment with endpoints on a parabola is a chord of the parabola. The value |4p| is the focal width of the parabola – the length of the chord through the focus and perpendicular to the axis.

8. Parabolas with vertex (0,0)

9. Example 1a Find the focus, the directrix, and focal width of the parabola x2 = -12y x2 = 4(-3)y The focus is (0, -3) The directrix is y = 3 The focal width is |4(-3)|=12

10. Example 1b Find the focus, the directrix, and focal width of the parabola x = 2y2 y2 = ½ x The focus is (1/8,0) The directrix is x = –1/8 The focal width is |4(1/8)| = ½

11. Example 2 Find the equation in standard form for a parabola whose a) directrix is the line x = 5 and focus is the point (-5, 0) standard form is y2 = 4px with p = -5 y2 = 4(-5)x y2 = -20x b) directrix is the line y =6 and vertex is (0,0) standard form is x2 = 4py with p = -6 x2 = 4(-6)y x2 = -24y

12. Example 3 Find the equation in standard form for a parabola whose a) vertex is (0,0) and focus is (0, -4) standard form is x2 = 4py with p = -4 x2 = 4(-4)y x2 = -16y b) vertex is (0,0), opens to the left with focal width 7 standard form is y2 = 4px with 4p = -7 y2 = -7x

13. Homework Page 641: 1,2, 5-20