# Why adding polynomial terms make linear function non-linear?

I would love to know why polynomial terms could make non-linear functions. From my understanding, it is just about using the current independent variables, so the relationship between $$x$$ and $$y$$ should be still linear?

A function $$f: \mathbb{R}^m \rightarrow \mathbb{R}$$ is called a linear function if it possesses the following "linearity" properties:

$$\begin{matrix} f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) & & \text{for all } \mathbf{x}, \mathbf{y} \in \mathbb{R}^m, \\[6pt] f(a\mathbf{x}) = a f(\mathbf{x}) \quad \ & & \quad \quad \quad \ \ \text{for all } \mathbf{x} \in \mathbb{R}^m \text{ and } a \in \mathbb{R}, \\[6pt] \end{matrix}$$

Now, consider the following polynomial regression function:

$$u(x, \boldsymbol{\beta}) = \beta_0 + \beta_1 x + \beta_2 x^2 + \cdots + \beta_k x^k.$$

If you apply the above definitions you can easily show that $$u$$ is linear with respect to the parameter vector $$\boldsymbol{\beta}$$ but not with respect to the value $$x$$. (As an exercise, see if you can show that the first linearity equation does not hold for this latter variable.) Now, in the context of regression models we call the model "linear" if it is linear with respect to the parameters rather than the explanatory variables (see related question here), so we would still call this a linear model. But this does not change the fact that the regression function is not a linear function of $$x$$.

If you are just sticking with your values of $$X$$ and consider these particular measurements to be your main point of interest, then I see how you can interpret the relationship between $$X$$ and $$Y$$ to be linear.

However, most of the times we are interested not in the measurements themselvels, but what they represent.

For example, consider a very simple case of predicting wage with age. We are interested in the relationship between these two variables, and if the relationship would be linear, then the older the person is, the more we would predict that they earn. But even intuitively this is not the case. We would expect the wage to be decreasing with age after some point (for example, after age = 60). That means that, in order to capture this relationship, we can introduce higher degrees of age to the model. Yes, we add more "independent variables" to the model, but the relationship that we are interested in is still between the age and the wage.