SOLUTION: what is the cordinates of the vertext in the parabola x^2+2x-8y+1=0

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Question 87090: what is the cordinates of the vertext in the parabola x^2+2x-8y+1=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B2x-8y%2B1=0 Start with the given equation

x%5E2%2B2x-8y=-1 Subtract 1 from both sides

2x-8y=-1-x%5E2 Subtract x%5E2 from both sides

8y=-1-x%5E2-2x Subtract 2x from both sides

8y=-x%5E2-2x-1 Rearrange the terms

y=%28-x%5E2-2x-1%29%2F-8 Divide both sides by -8

y=%281%2F8%29x%5E2%2B%282%2F8%29x%2B%281%29%2F8 Break up the fraction

y=%281%2F8%29x%5E2%2B%281%2F4%29x%2B%281%29%2F8 Reduce the middle term

Now lets complete the square to get the quadratic into vertex form

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=%281%2F8%29+x%5E2%2B%281%2F4%29+x%2B1%2F8 Start with the given equation



y-1%2F8=%281%2F8%29+x%5E2%2B%281%2F4%29+x Subtract 1%2F8 from both sides



y-1%2F8=%281%2F8%29%28x%5E2%2B2x%29 Factor out the leading coefficient %281%2F8%29



Take half of the x coefficient 2 to get 1 (ie %281%2F2%29%282%29=1).


Now square 1 to get 1 (ie %281%29%5E2=%281%29%281%29=1)





y-1%2F8=%281%2F8%29%28x%5E2%2B2x%2B1-1%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1 does not change the equation




y-1%2F8=%281%2F8%29%28%28x%2B1%29%5E2-1%29 Now factor x%5E2%2B2x%2B1 to get %28x%2B1%29%5E2



y-1%2F8=%281%2F8%29%28x%2B1%29%5E2-%281%2F8%29%281%29 Distribute



y-1%2F8=%281%2F8%29%28x%2B1%29%5E2-1%2F8 Multiply



y=%281%2F8%29%28x%2B1%29%5E2-1%2F8%2B1%2F8 Now add 1%2F8 to both sides to isolate y



y=%281%2F8%29%28x%2B1%29%5E2%2B0 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1%2F8, h=-1, and k=0. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=%281%2F8%29x%5E2%2B%281%2F4%29x%2B1%2F8 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%281%2F8%29x%5E2%2B%281%2F4%29x%2B1%2F8%29 Graph of y=%281%2F8%29x%5E2%2B%281%2F4%29x%2B1%2F8. Notice how the vertex is (-1,0).



Notice if we graph the final equation y=%281%2F8%29%28x%2B1%29%5E2%2B0 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%281%2F8%29%28x%2B1%29%5E2%2B0%29 Graph of y=%281%2F8%29%28x%2B1%29%5E2%2B0. Notice how the vertex is also (-1,0).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.






So the vertex is (-1,0)