Parabolic reflectors with microphones located at the focus are used to capture sounds from a distance. The sound waves enter the reflector and are concentrated toward the microphone. Write an equation to model a different parabolic reflector that is 4 feet wide and 2 feet deep, if the vertex of the reflector is located at (3,5) and the parabola opens to the left.

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Kelsie Marks · Accepted Answer

The parabola is horizontal so the standard form is:  (  y  −  k      )          2        =  4  p  (  x  −  h  )We are given that the vertex is at (3,5), then h=3 and k=5:  (  y  −  5      )          2        =  4  p  (  x  −  3  )To find p, substitute a point. A point on the parabola is   (  3  −  2  ,  5  −      4    2    )  =  (  1  ,  3  ) since the depth is 2 ft and the width is 4 m (half of it), which are subtracted from the coordinates of the vertex since it opens left Hence,  (  3  −  5      )          2        =  4  p  (  1  −  3  )4=-8p-0.5=pSo, the equation is:  (  y  −  5      )          2        =  4  (  −  0.5  )  (  x  −  3  )  (  y  −  5      )          2        =  −  2  (  x  −  3  )

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