# Linear Regression Model, linearity in parameters/ variables

I am confusing with the wording here. I was reading a book on linear regression. "The primary concern for linear models is that they display linearity in the parameters. Therefore, when we refer to a linear regression model we generally assume that the equation is linear in the parameters; it may or may not be linear in the variables"

Can anyone explain that last sentence to me? As it seems a bit odd to say so.. Like: y=a+b*x^2 In my understanding, this is linear in parameters, not linear in the variables, so that makes it a linear model? It feels like it should be the other way round, can someone please explain? Thanks.

Lets take an example $$y = \sum_{k=0}^n a_k x^k$$ where the first few terms are $$y = a_0 + a_1 x + a_2 x^2 +\cdots a_nx^n$$ we can re-label things as $v_k = x^k$ we have $$y = a_0 + a_1 v_1 + a_2 v_2 + \cdots + a_nv_n$$ where we regress on the different variables $v_k$'s or linear in parameters. A nonlinear function which is not linear in parameters are given by $$y = \frac{x}{a_0+a_1x}$$ or $$y = \sqrt{a_0 + a_1 x+a_2x^2}$$ etc..