SOLUTION: Find the vertex, focus, directrix, and axis of symmetry of each parabola. x^2+8y+4x-4=0 We have been using the formulas y-k=a(x-h)^2 and x-h=a(y-k)^2 I don't understand which to

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Question 193521This question is from textbook algebra and trigonometry structure and method book 2
: Find the vertex, focus, directrix, and axis of symmetry of each parabola.
x^2+8y+4x-4=0
We have been using the formulas y-k=a(x-h)^2 and x-h=a(y-k)^2 I don't understand which to use here, even after completing the square and its confusing because I don't know which is which. please explain and be specific. thank you.This question is from textbook algebra and trigonometry structure and method book 2

Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B8y%2B4x-4=0 Start with the given equation.


x%5E2%2B4x%2B8y-4=0 Rearrange the terms.


Now we need to complete the square for the "x" terms.


Take half of the x coefficient 4 to get 2. Square 2 to get 4.


x%5E2%2B4x%2B4%2B8y-4=4 Add the previous result 4 to both sides.


x%5E2%2B4x%2B4%2B8y-4-4=0 Subtract 4 from both sides.


x%5E2%2B4x%2B4%2B8y-8=0 Combine like terms.


%28x%2B2%29%5E2%2B8y-8=0 Factor x%5E2%2B4x%2B4 to get %28x%2B2%29%5E2


8y-8=-%28x%2B2%29%5E2 Subtract %28x%2B2%29%5E2 from both sides


8%28y-1%29=-%28x%2B2%29%5E2 Factor out the GCF 8 from the left side


y-1=-%281%2F8%29%28x%2B2%29%5E2 Multiply both sides by 1%2F8


y-1=-%281%2F8%29%28x-%28-2%29%29%5E2 Rewrite x%2B2 as x-%28-2%29



Now the equation y-1=-%281%2F8%29%28x-%28-2%29%29%5E2 is in the form y-k=a%28x-h%29%5E2 where a=-1%2F8, h=-2 and k=1


Vertex:

The vertex of y-k=a%28x-h%29%5E2 is (h,k). Since h=-2 and k=1, the vertex of y-1=-%281%2F8%29%28x-%28-2%29%29%5E2 is (-2,1)


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Focus:

First, we define p=1%2F%284a%29 where the absolute value of "p" is the distance from the vertex to the focus.

Note: the formula p=1%2F%284a%29 is derived from the equation %28x-h%29%5E2=4p%28y-k%29


The focus of y-k=a%28x-h%29%5E2 is the point . In other words, simply add the value of "p" to the y coordinate of the vertex to get the focus.


So let's find "p": p=1%2F%284a%29=1%2F%284%28-1%2F8%29%29=1%2F%28-1%2F2%29=-2. So p=-2


Since h=-2, k=1, and p=-2, this means that k%2Bp=1-2=-1


So the focus of the form is the point (-2,-1)


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Directrix:



The directrix of y-k=a%28x-h%29%5E2 is the equation y=k-p.


Since y=k-p=1-%28-2%29=3, this means that the directrix is y=3


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Axis of Symmetry:


In this case, the axis of symmetry is simply the equation of the vertical line through the vertex.


Since h=-2, this means that the axis of symmetry is x=-2



Visual Check:

drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1%2F8%29%28x%2B2%29%5E2%2B1%29%2C%0D%0Agreen%28line%28-20%2C3%2C20%2C3%29%29%2C%0D%0Ablue%28line%28-2%2C-20%2C-2%2C20%29%29%2C%0D%0Acircle%28-2%2C1%2C0.08%29%2Ccircle%28-2%2C1%2C0.10%29%2Ccircle%28-2%2C1%2C0.12%29%2C%0D%0Acircle%28-2%2C-1%2C0.08%29%2Ccircle%28-2%2C-1%2C0.10%29%2Ccircle%28-2%2C-1%2C0.12%29%0D%0A%29


Graph of x%5E2%2B8y%2B4x-4=0 with the vertex, focus, directrix (green) and axis of symmetry (blue)