The basic form is what you could call the shape of the quadratic. All quadratics look like U or an inverted U, without reference to the spread of the arms of the U or where it sits in the reference frame. They’re all parabolas.
The basic form is in an “obvious” position and is an upright U. The most obvious position is a seat at the origin (0,0), which would be the lowest point of the U. The equation is y=x².
The basic form is sometimes called the “parent” and its transformations are its “children”. Transformations can be a “seat” shift. The seat is called the vertex, and it can be shifted up, down, left or right. It can also be dilated—magnified or reduced—and reflected (turned upside down). So the shift transformation moves the vertex from (0,0) to some other coordinate which is often represented by (h,k), displacement of x by the amount h and displacement of y by the amount k. The dilation factor is usually represented by a, which can be positive or negative. Negative values produce an inverted U shape, or reflection of the basic shape. So a transformed parabola has the equation: y-k=a(x-h)². When we expand this we get y=ax²-2ahx+h²+k. That’s the quadratic. The constant in a quadratic y=ax²+bx+c is equal to h²+k and b=-2ah, so it’s possible to work out a, h and k from the coefficients in the quadratic. Knowing the vertex and y intercept (= constant c or h²+k) helps to plot the parabola. The x intercepts (if they exist) are the zeroes of the quadratic. Not all parabolas will have x intercepts, but they all have a y intercept.
(Parabolas lying on their side have the basic form x=y².)