The parameters of the quadratic regression model can be confusing. This applet depicts a quadratic regression model, which is equivalent to an interaction model in which the predictor variable X interacts with itself. That is the relationship between X and Y depends on the level of X.

The applet illustrates which parameters in the model do or do not depend on the zero point of the X predictor variable. Drag the vertical axis to create a new predictor variable X', which equals the original the predictor X with a constant subtracted. The zero points on the X and X' axes do not correspond. As you drag the vertical axis, observe the changes in the model coefficients.

At any given value of X, the simple relationship between X and Y is depicted by the red
tangent line. Drawing the tangent line at the point X' = 0 illustrates the meaning of
the parameters. In particular, the tangent line at X' = 0 is defined by

Y = b0' + b1' X

The coefficient for X' (depicted in red)
necessarily changes as the zero point for X' changes. Similarly, the
intercept--the actual tangent point at the Y axis--necessarily
changes as the zero point for X' changes.
Finally, the coefficient for the product (depicted in black),
representing the quadratic effect, is invariant
and represents half the amount the tangent slope changes as X chagnes by one unit.