Algebra 2 Calculating a Parabola Focus and Directrix
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Focus on Parabolas
Calculating a Parabola Focus and Directrix

 

 Parabolas do have focus points,

 but since a parabola only has one "U,"
 it only has one focus point.
 
 Example:
 Find the focus point on the parabola:

 

 Y = 2X2

 

 

 
 There are two steps to the process.
 The first is to move the coefficients to the Y
 The second step is to factor a 4 out of the Y coefficient
 (who knows why a 4 works for this, but it does)
 What's left as the Y coefficient is the distance
 
 So:
 Step 1

 

 
 Step 2:

 

 
 So the distance from the bottom of the "U" of the parabola
 to the focus point is 1/8 unit.
 

 

 
 Let's try another one ...
 
 Example:
 Find the focus point of the equation:

 Y = 8(X - 3)2 + 2

 

 

 
 Step 1:
 Move the 8 away from the X 2 term
 

 

 
 Step 2:
 Factor the mysterious 4 out of the Y coefficient
 

 

 
 The distance from the bottom of the "U" (called the vertex)
 to the focus point is 1/32.
 

 

 
 So what does this focus point do?
 
 By itself, not much.
 But let's make it do something of cosmic significance.
 OK, maybe it's not THAT big a deal ...
 
 Draw a line below the vertex of the parabola 
 the same distance as the distance to the focus.
 

 

 
 Here's the deal. 
 From any point on the parabola,
 it's the same distance to the focus point
 as it is to the closest point on the line we just drew ...
 

 

 
 OK, OK, this seemed like a real yawner to me when I learned it too,
 but there's a bunch of stuff in Physics that actually uses these things.
 Those tricks are enough to learn at one time
 so we show you the math part now.
 Then when you get to the Physics stuff, it isn't all
 hitting you at one time.
 
 Not going on to physics you say?
 Don't care about math and science?
 
 OK, No problem.
 You can practice for a career without them 
 right here and now.
 
 Just repeat after me:
 
 "Do you want ..."
 
 "fries with that?"
 
 Still with me?
 OK, back to parabolas ...
 
 If the X2 term has a minus sign in front of it
 the "U" is upside down.
 That would make the focus point below the parabola vertex
 and the horizontal line above it ...
 

 

 
 But everything would still work the same.
 
 We could even reverse the X and Y.
 
 Example:
 Find the focus point of:

 X = 2Y2

 

 

 
 Since this one is "sideways," 
 move the coefficient TO the X term
 and factor out the magic 4.
 

 

 
 So the focus point is 1/8 unit to the right
 of the vertex of the "U"
 And the line is 1/8 unit to the left
 of the vertex of the "U."
 

 

 
 Math types didn't like calling the line "The Line."
 They like fancy names that sound important.
 So they came up with a fancy name for the line.
 They call it:

 THE DIRECTRIX

 
 Whatever!
 

   copyright 2005 Bruce Kirkpatrick

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