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Need some help with this one please!

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85
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What is the equation of a parabola with (−2, 4) as its focus and y = 6 as its directrix?

Guest Jun 15, 2017

#2
+806
+1

There's an excellent formula that I concocted 5 days ago to always give the equation of a parabola when only the directrix and focus is given. Here is the equation:

Let a = x-coordinate of focus

Let b = y-coordinate of focus

Let k= equation of line of the directrix

$$y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}$$

All you have to do is plug into this formula, simplify, and you're done. Let's try this together:

a=-2

b=4

k=6

 $$y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}$$ Plug in the appropriate values that are given by the focus and directrix. $$y=\frac{(x-(-2))^2}{2(4-6)}+\frac{4+6}{2}$$ Let's clean this up a bit, shall we? $$y=\frac{(x+2)^2}{-4}+5$$ Technicaly, you could stop here and call it a day, but I am going to attempt to make it look even cleaner! I'll expand the $$(x+2)^2$$ $$y=-\frac{x^2+4x+4}{4}+5$$ After expanding it, I'll change 5 into an improper fraction that I can add to the current fraction. $$y=-\frac{x^2+4x+4}{4}+\frac{20}{4}$$ Since the fractions have common denominators. Add the fractions together, but you have to be very attentive to how you do this. Notice how there is a negative. I'm going to get rid of this because you can't combine a negative fraction with a positive one; it just doesn't work. $$y=\frac{-(x^2+4x+4)}{4}+\frac{20}{4}$$ I'm distributing that negative because you cannot combine a negative and positive fraction. $$y=\frac{-x^2-4x-4}{4}+\frac{20}{4}$$ Now, you can add the fractions together. Now you add the fractions together normally. $$y=\frac{-x^2-4x+16}{4}$$ Break off the two last terms from the current fraction. You'll see why. $$y=\frac{-x^2}{4}+\frac{-4x+16}{4}$$ The rightmost fraction can be simplified because a factor of 4 goes into both terms. Wow! $$y=\frac{-x^2}{4}-x+4$$ $$y=-\frac{1}{4}x^2-x+4$$ This is your final equation.

No, I did not just pull that formula out of the air! I derived it myself. If you are wondering why this formula will work 100% of the time, I explained it, in detail, here where someone asked a similar question to yours.

https://web2.0calc.com/questions/parabola-equation#r3

TheXSquaredFactor  Jun 15, 2017
Sort:

#1
+75298
+1

We want the form

4p (y - k)  =  (x - h)^2

Where  the vertex is (-2 ,6+ p)    and  p = ( y coordinate of the focus - y value of the directrix) / 2 =( 4 - 6)/2  = -2/2  = -1

So the vertex  = (-2, 5)  = (h,k)

So we have

4(-1) (y -5)  = ( x - -2)^2    simplify

-4(y - 5) = ( x + 2)^2

-4y + 20  =  x^2 + 4x + 4   subtract 20 from both sides

-4y  = x^2 + 4x - 16          divide through  by  -4

y = (-1/4)x^2  - x  + 4

Here's the graph : https://www.desmos.com/calculator/1khnfskw3p

CPhill  Jun 15, 2017
#2
+806
+1

There's an excellent formula that I concocted 5 days ago to always give the equation of a parabola when only the directrix and focus is given. Here is the equation:

Let a = x-coordinate of focus

Let b = y-coordinate of focus

Let k= equation of line of the directrix

$$y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}$$

All you have to do is plug into this formula, simplify, and you're done. Let's try this together:

a=-2

b=4

k=6

 $$y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}$$ Plug in the appropriate values that are given by the focus and directrix. $$y=\frac{(x-(-2))^2}{2(4-6)}+\frac{4+6}{2}$$ Let's clean this up a bit, shall we? $$y=\frac{(x+2)^2}{-4}+5$$ Technicaly, you could stop here and call it a day, but I am going to attempt to make it look even cleaner! I'll expand the $$(x+2)^2$$ $$y=-\frac{x^2+4x+4}{4}+5$$ After expanding it, I'll change 5 into an improper fraction that I can add to the current fraction. $$y=-\frac{x^2+4x+4}{4}+\frac{20}{4}$$ Since the fractions have common denominators. Add the fractions together, but you have to be very attentive to how you do this. Notice how there is a negative. I'm going to get rid of this because you can't combine a negative fraction with a positive one; it just doesn't work. $$y=\frac{-(x^2+4x+4)}{4}+\frac{20}{4}$$ I'm distributing that negative because you cannot combine a negative and positive fraction. $$y=\frac{-x^2-4x-4}{4}+\frac{20}{4}$$ Now, you can add the fractions together. Now you add the fractions together normally. $$y=\frac{-x^2-4x+16}{4}$$ Break off the two last terms from the current fraction. You'll see why. $$y=\frac{-x^2}{4}+\frac{-4x+16}{4}$$ The rightmost fraction can be simplified because a factor of 4 goes into both terms. Wow! $$y=\frac{-x^2}{4}-x+4$$ $$y=-\frac{1}{4}x^2-x+4$$ This is your final equation.

No, I did not just pull that formula out of the air! I derived it myself. If you are wondering why this formula will work 100% of the time, I explained it, in detail, here where someone asked a similar question to yours.

https://web2.0calc.com/questions/parabola-equation#r3

TheXSquaredFactor  Jun 15, 2017

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