this is saying you can ge the vertex you can out it into common for and then the h and k will be equal to the vertex.

this is saying that to make the parabola narrower the coeficent increases and to make it wider it must become a decimal number but to keep it in the standard form the coeficent needs to be 1

This is the common form of all quadratic functions similar to y=mx+b is to linear functions.