I've recently been re-acquainting myself with linear algebra and can admit that econometrics isn't my best subject. I'm trying to prove the existence of a "Kuznet's Curve" relationship between my regressors and regressand. The model uses panel data (cross sectional, time components). An earlier study models the relationship as:

(1) yit = b1 + b2.xit + b3.xit2 + b4.zit + uit

where x = IV, with k1 x 1 matrix,
z = IV, with k2 x 1 matrix
u(t) = error term, normally distributed, mean 0, variance of o^2.

My questions are:

1. Does the above model represent a bi-variate quadratic equation?
- various sites list the general form of a bi-variate quadratic as (2)

- My assumption is that equation 1 is obviously a bi-variate quadratic, much in the same way that y=x2 is still a quadratic equation (general form of y=x + x2 +c)

2. Are the illustrations of bi-variate quadratics correct in the below link?
Curvilinear Regression

Or more importantly, the b4 coefficient of equation 1 has a linear relationship with y, and will affect the steepness of the curve.

I think the answers are pretty obvious (yes it is a bi-variate quadratic), but lack of sleep has made me illogical made me-over think things! Would appreciate the help!!