OHHS Pre-Calculus Mr. J. Focht Analytic Geometry in Two and Three Dimensions Chapter 8 OHHS Pre-Calculus Mr. J. Focht

8.1 Conic Section: the Parabola What You'll Learn Geometry of a Parabola Translations of Parabolas Reflective Property 8.1

Parabola Set of all points that are equidistant from a line and a point. Directrix Focus Vertex Axis of Symmetry Any point (x,y) is as far from the line as it is from the focus 8.1

A Parabola is a Conic Section 8.1

Parabola All parallel beams reflect through the focus Why is this important? Think satellite dish, flashlight, headlight 8.1

Parabola Equation Vertex (0,0) F(0, p) p A(x,y) p D(x,-p) p = Focal Length = distance from focus to vertex 8.1

Parabola Equation Vertex (0,0) A(x,y) p F(0, p) D(x,-p) 8.1

Parabola Equation Vertex (0,0) 8.1

Parabola Equation Vertex (0,0) 8.1

Definition: Latus Rectum Segment passing through the focus parallel to the directrix Focal Width is the length of the Latus Rectum. This length is |4p| 4p 8.1

Parabola Equations Summary x2 = 4py p > 0 x2 = 4py p < 0 y2 = 4px p > 0 y2 = 4px p < 0 8.1

Example Find the focus, the directrix, and the focal width of the parabola y = -½x2. First put into standard form x2 = -2y 4p = -2 p = -½ D(0,½) Y = ½ (0,0) FW = 2 F(0,-½) 8.1

Now You Try P. 641, #1 : Find the focus, the directrix, and the focal width of the parabola x2 = 6y 8.1

Example Find an equation in standard form for the parabola whose directrix is the line x=2 and whose focus is the point (-2,0). y2 = 4px y2 = -8x x=2 (-2,0) (0,0) p = -2 p = -2 8.1

Now You Try P. 641 #15: Find the standard form of a parabola with focus (0, 5), directrix y=-5 8.1

Standard Form of the Equation of Vertex (h, k) (y-k)2 = 4p(x-h) p Directrix (h, k) 8.1

Standard Form of the Equation of Vertex (h, k) (x-h)2 = 4p(y-k) p (h, k) Directrix 8.1

Example (y-3)2 = 8(x+4) Find the equation of this parabola. (y-k)2 = 4p(x-h) p = distance from vertex to focus (-4,3) (-2,3) p = 2 h = -4 k = 3 (y-3)2 = 8(x+4) 8.1

Example (y+1)2 = -10(x-3) Find the equation of this parabola. (y-k)2 = 4p(x-h) p = distance from vertex to focus (-2, -1) (3, -1) p = -5 h = 3 k = -1 (y+1)2 = -10(x-3) 8.1

Now You Try P. 641, #21 Find the equation of this parabola. Focus (-2, -4), vertex (-4, -4) 8.1

What’s the sign of p? p > 0 p < 0 8.1

Graphing a Parabola Graph (y-4)2 = 8(x-3) Change to 8.1

Example x2-6x-12y-15=0 x2 -6x = 12y + 15 h = 3 k = -2 p = 3 Find the coordinates of the vertex and focus, and the equations of the directrix and axis of symmetry. x2-6x-12y-15=0 Put the equation into standard form. x2 -6x = 12y + 15 + 9 + 9 (x-3)2 = 12y+24 (x-3)2 = 12(y+2) h = 3 k = -2 p = 3 8.1

Example (x-3)2 = 12(y+2) h = 3 k = -2 p = 3 x = 3 (3, 1) (3, -2) The focus is 3 above the vertex. y = -5 The directrix is a horizontal line 3 below the vertex. (3, -5) The line of symmetry passes through the vertex and focus 8.1

Now Your Turn P. 651 #49: Prove that the graph of the equation x2 + 2x – y + 3 = 0 is a parabola, and find its vertex, focus, and directrix. 8.1

Sketch a Graph Example Graph (y-2)2 = 8(x-1) The vertex is (1,2) Focus = (3,2) Focal Width = 8 4 above & 4 below the focus. (3,6) (1,2) (3,2) (3,-2) 8.1

Now Your Turn P. 641, #33: Sketch the parabola by hand: (x+4)2 = -12(y+1) 8.1

Application Example On the sidelines of each of its televised football games, the FBTV network uses a parabolic reflector with a microphone at the reflector’s focus to capture the conversations among players on the field. If the parabolic reflector is 3 ft across and 1 ft deep, where should the microphone be placed? 8.1

Application Example x2 = 4py (1.5, 1) is on the parabola. p = 0.5625 ft p = 6.75 in The microphone should be placed inside the reflector along its axis and 6.75 inches from its vertex. 8.1

Your Turn Now P. 652, #59 The mirror of a flashlight is a paraboloid of revolution. Its diameter is 6 cm and its depth is 2 cm. How far from the vertex should the filament of the light bulb be placed for the flashlight to have its beam run parallel to the axis of its mirror? 8.1

Home Work P. 641-642 #2, 4, 12, 16, 22, 28, 34, 42, 50, 52, 56, 60, 65-70 8.1

The End