Still don't know the reason why is vertex form needed from the first place

[Gazoon007]

I’ve watched it a couple of times, but I still don’t get the idea of how the different forms of the equation could determine the vertex by changing it to vertex form. How could the first mathematician who have found the idea to know the middle point of parabola’s (h) and the vertex along the y-axis (k) from the standard form by changing it to vertex form?

You had given us an analogy of how it changes the standard to vertex form by graphically by drawing a square, it helps to know in the terms of algebra calculation, but it lacks the reason why is vertex form needed from the first place.

Sorry if I overcomplicate stuff, I just want to know the reason behind all things in math so I can get strong retention of the concept.

[garypettie]

Hi @Gazoon007, hopefully this explanation will answer your question as to why vertex (or squared) form actually provides us with the vertex of the parabola.

So we’re working with the equation y = a(x - h)^2 + k.
And we’ll assume that a has some positive value for this example.

Let’s start by thinking about the term (x - h)^2.
Our first observation is that (x - h) is squared, so this term can only ever be positive or zero.

Now, at the point where (x - h)^2 equals zero (the smallest value it can be) the rest of our equation will cancel down to just y = k.
And if (x - h)^2 equals zero, this means that x = h.

Recall that, when a is positive, the lowest point on the parabola is the vertex.
And we just found that the lowest possible values for our equation occur when x = h and y = k.
Therefore, the coordinates for the vertex must be (h, k).

You can also follow similar logic for n-shaped parabolas, but hopefully this gives you an idea of why vertex form works the way that it does.

If you have any other questions, please do let me know.