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**OHHS Pre-Calculus Mr. J. Focht**

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

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**8.1 Conic Section: the Parabola**

What You'll Learn Geometry of a Parabola Translations of Parabolas Reflective Property 8.1

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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

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**A Parabola is a Conic Section**

8.1

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**Parabola All parallel beams reflect through the focus**

Why is this important? Think satellite dish, flashlight, headlight 8.1

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**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

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**Parabola Equation Vertex (0,0)**

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

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**Parabola Equation Vertex (0,0)**

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**Parabola Equation Vertex (0,0)**

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**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

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**Parabola Equations Summary**

x2 = 4py p > 0 x2 = 4py p < 0 y2 = 4px p > 0 y2 = 4px p < 0 8.1

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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

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

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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

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

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**Standard Form of the Equation of Vertex (h, k)**

(y-k)2 = 4p(x-h) p Directrix (h, k) 8.1

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**Standard Form of the Equation of Vertex (h, k)**

(x-h)2 = 4p(y-k) p (h, k) Directrix 8.1

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**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 = k = 3 (y-3)2 = 8(x+4) 8.1

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**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 = k = -1 (y+1)2 = -10(x-3) 8.1

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**Now You Try P. 641, #21 Find the equation of this parabola.**

Focus (-2, -4), vertex (-4, -4) 8.1

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What’s the sign of p? p > 0 p < 0 8.1

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

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**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

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**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

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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

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**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

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

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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

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**Application Example x2 = 4py (1.5, 1) is on the parabola.**

p = 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

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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

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

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The End

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