SOLUTION: I am having trouble finding the vertex of parabolas. I need to find the axes, roots, and the vertices of the 3 following questions. Can you please show it to me in the completing t

Question 183003: I am having trouble finding the vertex of parabolas. I need to find the axes, roots, and the vertices of the 3 following questions. Can you please show it to me in the completing the square method.
The questions are
y=2x^2-x+6
y=x^2-8x+2
y=4x^2-x+1
Thanks in advance.
Found 2 solutions by stanbon, josmiceli:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
I need to find the axes, roots, and the vertices of the 3 following questions. Can you please show it to me in the completing the square method.
The questions are
y=2x^2-x+6
2x^2 - x + ? = y-6+?
2(x^2 - (1/2)x + (1/4)^2 = y - 6 + 2(1/4)^2
2(x- (1/4))^2 = y - (48/8) + (1/8)
2(x-(1/4)^2 = y - (47/8)
(x - (1/4)^2 = (1/2)(y -(47/8))
------------------------------------
Vertex: (1/4 , (-47/8))
vertical axis: x = 1/4
Roots: No Real roots
Use Quadratic formula to find the complex roots.
-----------------------------------
y=x^2-8x+2
x^2 - 8x = y-2
x^2 - 8x + 16 = y -2 + 16
(x-4)^2 = y + 14
----------------------
Vertex: (4 , -14)
vertecal axis: x = 4
Roots: Use the quadratic formula.
---------------------------
y=4x^2-x+1
4x^2 - x = y - 1
4(x^2 - (1/4)x + (1/8)^2) = y - 1 + 4*(1/8)^2
4(x - (1/8))^2 = y - 15/16
(x-(1/8))^2 = (1/4)(y - 15/16)
----
Vertex: (1/8 , 15/16)
vertical axis: x = 1/8
Roots: Use the Quadratic formula
=====================================
Cheers,
Stan H.

Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
The x-co-ordinate, , of the vertex is exactly midway
between the roots, so if the roots are and
the vertex is at (,)
Another way to find the vertex is, if the equation is in
the form , then is at
. I'll solve both ways:
(1)
To complete the square, take 1/2 the co-efficient of ,
square it, and add it to both sides.
First set equation equal to and subtract
from both sides

Divide both sides by

Take the square root of both sides

The 2 roots are:

and

Now I find the point midway between the roots

(notice the terms with cancel)

And now the easy way:

Now just plug this into the equation to find

I'll plot it

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