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**Reflective Property of Parabolas**

One more day in Sec. 8.1b…

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**But first, let’s start with a problem:**

Prove that the graph of the given equation is a parabola, and find its vertex, focus, and directrix. Write in standard form: V: (1, 7/6), F: (1, 5/3), D: y = 2/3

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**More Practice V: (13/4, 1), F: (9/4, 1), D: x = 17/4**

Prove that the graph of the given equation is a parabola, and find its vertex, focus, and directrix. Write in standard form: V: (13/4, 1), F: (9/4, 1), D: x = 17/4

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**More Practice Write an equation for the parabola graphed below.**

General equation: Substitute in vertex: (11/2,0) Substitute in point, solve for p: V(1,–3) Final Answer:

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**Reflective Property of Parabolas**

If a parabola is rotated in 3-D space around its axis, the parabola “sweeps out” a paraboloid of revolution. These are important because a signal (light, sound, radio waves, etc.) emanating from the focus reflects off the surface of the dish in lines parallel to the axis (and vice versa – incoming parallel waves are sent directly toward the focus…). Check out Figure 8.9 on p.637!!!

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Practice Problems During football games, a TV 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? General equation: A diagram of a cross-section of the reflector: Substitute the points: (–1.5, 1) (1.5, 1) F(0, p) V(0, 0)

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Practice Problems During football games, a TV 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? A diagram of a cross-section of the reflector: Because p = ft, or 6.75 inches, the microphone should be placed inside the reflector along its axis and 6.75 inches from its vertex. (–1.5, 1) (1.5, 1) F(0, p) V(0, 0)

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**Practice Problems Because p = 1.125 cm, the filament should be placed**

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? Because p = cm, the filament should be placed 1.125 cm from the vertex along the axis of the mirror.

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**Practice Problems Lengths (in feet): 79.444, 54.444, 35, 21.111,**

A main cables of a suspension bridge uniformly distribute the weight of the bridge when in the form of a parabola. The main cables of a particular bridge are attached to towers that are 600 ft apart. The cables are attached to the towers at a height of 110 ft above the roadway and are 10 ft above the roadway at their lowest points. If vertical support cables are at 50-ft intervals along the level roadway, what are the lengths of these vertical cables? Lengths (in feet): , , 35, , 12.778, 10, , , 35, ,

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