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Question 967250: Can someone pretty please help me figure this out?
For the following equation of a parabola determine the vertex, focus, and directix, then sketch it's graph.
x^2+4x-3y+4=0
Thank you so much!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
try this reference.
http://www.purplemath.com/modules/parabola.htm
the information pertinent to your problem is as follows:
you want to know the vertex, the focus and the directrix.
there are 2 basic forms of the equation of a parabola that you will need.
the first form is the vertex form.
that form is y = a * (x-h)^2 + k
(h,k) is the vertex.
a is the coefficient of the (x-h)^2 term.
you need to transform your equation to this form.
start with:
x^2+4x-3y+4=0
associate the first two terms together and then complete the squares on them.
you will get:
(x^2 + 4x) - 3y + 4 = 0.
complete the squares on the x terms to get:
(x + 2)^2 - 4 - 3y + 4 = 0
combine like terms to get:
(x + 2)^2 - 3y = 0
add 3y to both sides of the equation to get:
(x + 2)^2 = 3y
divide both sides of the equation by 3 to get:
(x + 2)^2 / 3 = y
this is the same as y = 1/3 * (x + 2)^2
this is now in the vertex form of y = a * (x-h)^2 + k
a = 1/3
h = -2
k = 0
the vertex is at (h,k) = (-2,0)
the second form you will need is the conics form of the equation of a parabola.
that form is 4p(y – k) = (x – h)^2
start with the vertex form of y = 1/3 * (x + 2)^2
multiply both sides of the equation by 3 to get:
3y = (x + 2)^2
the equation is now in the conics form of 4p * (y - k) = (x - h)^2
this is because 3y is the same as 3 * (y - 0).
that's where the k of 0 comes from.
in this equation:
4p = 3
k = 0
h = -2
since 4p = 3, then divide by 4 to get p = 3/4.
p is the distance from the focus to the vertex and from the directrix to the vertex.
the vertex is in between the focus and the directrix.
the focus is inside the parabola.
the directrix is outside the partabola.
the vertex is at (-2,0)
the focus is at (-2,3/4)
the directrix is at y = -3/4
the directrix is a line perpendicular to the axis of symmetry of the parabola.
there is also the standard form of the equation of a parabola.
that form is y = ax^2 + bx + c.
you can convert your equation to this form by solving for y.
it's not necessary, however, to solve this problem.
solving this problem required the vertex form and the conics form.
the relationship between p in the conics form and a in the vertex form is:
4p = 1 / a
a = 1 / 4p
the vertex form of your equation was y = 1/3 * (x + 2)^2
the conics form of your equation was 3y = (x + 2)^2
4p is equal to 1/a which made 4p equal to 1 / (1/3) = 3.
a is equal to 1/(4p) which made a equal to 1/3.
the graph of your equation looks like this:
all 3 forms of the equation were graphed.
you can see that they all point to the same graph.
this means the different forms of the equation are equivalent to each other.
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