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the vertex of the parabola below is at the point (4, -1). Which could be this parabola's equation?

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4 Answers

The vertex of the parabola is at (4,-1).  Knowing the vertex alone will not determine a parabola.  Assume the parabola's equation is f(x) = ax2 + bx + c.
 
The x-coordinate of the vertex is x = -b/2a = 4, so that b = -8a.
 
(4,-1) is on the parabola, so f(4) = -1 = 16a + 4b + c;  
substituting, b = -8a,  -1 = 16a -32a + c, so that c = 16a -1.
 
So f(x) = ax2 -8ax + 16a - 1 = a(x2-8x+16) -1.   This is the FAMILY of parabolas with vertex (4,-1).   
 
The limit of the family as a approaches 0 is the horizontal line y = -1.   As a increases to infinity the parabola becomes steeper, until a approaches infinity, the parabola approaches the vertical line segment at x = 4 where y is in (-1,infinity).  As a decreases to -infinity, the parabola approaches the vertical like at x =4 where y is in the interval (-infinity, -1).
 
If we specify any point in the x-y plane where x is not equal to 4,  there is a unique parabola in this family that passes through that point.  
It's easy to see that y = a x^2 is a parabola with vertex at (0,0).
 
If it's translated by the vector <h,k> the equation becomes
(y - k) = a (x - h)^2, or y = a (x - h)^2 + k, and the vertex becomes (h,k).
 
(See sks23cu dot net slash MT slash Files slash Transformations slash sks23cuTransformations dot pdf)
 
If (h,k) = (4,-1), then the equation, by substitution, is y = a (x - 4)^2 - 1.
 
To find the value of "a" another point on the parabola must be used. Let's assume we know (5, -3) is a point on the parabola. Then:
-3 = a (5 - 4)^2 - 1 => -2 = a (1) => y = -2 (x - 4)^2 - 1.
The other way to approach this is to follow the general form of a parabola. Start with something simple.
y=x^2 is a parabola with a vertex at (0,0) right?
 
Well y=x^2 -1 would be a vertex at (0,-1). Getting close.
 
Now we need the vertex at x=4. So there's a neat trick here. I can prove it in a separate comment but it's easier to look at it logically. If y=(x-a)^2-1 would produce a vertex at y=-1, then (x-a)^2 would be 0. That means to have a vertex at x=4 would mean a=4.
 
So the equation could be y=(x-4)^2-1.
 
If you multiply it out, you'll find that this is x^2-8x+16-1 which is just what Parviz derived.
 
 

Comments

That is how the quadratic formula derives. Factoring with completing the square of aX2 + bx +c.

Comment

Vertex of a parabola with equation:
 
   f(x) = aX^2 + bX +c
 
  is at  Coordinate( ( -b/2a,b^2 - 4ac )
                                         4a^2      )
 
 f( x ) = k ( X - b/2a) ^2 - b^2 - 4ac
                                        4a^2
 
   For this case:
 
    f( x) = K( X - 4 ) ^2 - 1
            = k( X^2 -8X + 16 -1=
             = K ( X2- 8X +15) , have to have coordinate of another point of the curve to determine value of K,