Where is the vertex of the parabola f(x) = x^2 - 5x + 6?

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SUMMARY

The vertex of the parabola defined by the equation f(x) = x^2 - 5x + 6 is located at the point (5/2, -1/4). This conclusion is reached by recognizing that the x-coordinates of the roots, 2 and 3, indicate that the vertex lies halfway between these points. Additionally, the vertex can be confirmed by converting the equation into vertex form, y = a(x-h)^2 + k, where h and k represent the vertex coordinates. Completing the square also yields the same result, confirming the vertex's position.

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darkfire313
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See, there is an equation which is f(x) = x^2 - 5x + 6. On the points 2 and 3 of the x axis, they reach 0. So where is the vertex? I'm guessing its -.25, but just to make sure i had to ask. Well, our class seemed to have a lot of it and our teacher, Ms. Knudsvig, didn't know.
 
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darkfire313 said:
See, there is an equation which is f(x) = x^2 - 5x + 6. On the points 2 and 3 of the x axis, they reach 0. So where is the vertex? I'm guessing its -.25, but just to make sure i had to ask. Well, our class seemed to have a lot of it and our teacher, Ms. Knudsvig, didn't know.
You mean that f(2)= 4- 10+ 6= 0 and f(3)= 9- 15+ 6= 0. Because this parabola has vertical axis, it vertex is exactly half way between 2 and 3, at 5/2. f(5/2)= 25/4- 25/2+ 6= 25/4- 50/4+ 24/4= -1/4= 0.25.

You could also get that, without using the fact that the vertex is exactly halfway between to points of equal height, by completing the square:

5/2= 5/2 and (5/2)^2= 25/4 so f(x)= x^2- 5x+ 25/4- 25/4+ 6= (x- 5/2)^2- 25/4+ 24/4= (x- 5/2)^2- 1/4.

Since a square is never negative, that is always -1/4 plus something. When x= 5/2, (5/2- 5/2)^2- 1/4= -1/4. The lowest point, the vertex, is (5/2, -1/4).
 
darkfire313 said:
See, there is an equation which is f(x) = x^2 - 5x + 6. On the points 2 and 3 of the x axis, they reach 0. So where is the vertex?

To find the vertex, all you have to do is write the equation in vertex form. Standard from for a parabola is ax^2+bx+c. Vertex form for a parabola is y= a(x-h)^2+k. Once you have the equation in this form, h and k are the coordinates for your vertex.
 
How much do you (darkfire) know about calculus?
 

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