Notice that the $\hat{f}$ don't have an error term, $\epsilon$, as would be expected for regression models. Why don't these models have an error term? I am currently reading some notes on linear and polynomial regression. The notes say the following:

The linear model is
$$\hat{f}_L (X) = \beta_0 + \beta_1 X$$
The quadratic model is
$$\hat{f}_Q(X) = \beta_0 + \beta_1 X + \beta_2 X^2$$

Notice that the $\hat{f}$ don't have an error term, $\epsilon$, as would be expected for regression models. Why don't these models have an error term?

There are two ways to write the equation of a regression model. One describes the actual relationship between X and Y, and the other describes the estimated relationship.
This equation
$$ Y_i= \beta_0 + \beta_1 X_i +\epsilon_i$$
Means "For each observation "i" in the dataset, that observation's Y value equals (exactly!) $\beta_1$ multiplied by that observation's X value, plus  $\beta_0$, plus some value $\epsilon_i$." Note that this equation will always be true no matter how good or bad your model is.
By contrast, the equation
$$ \hat{Y_i}= \beta_0 + \beta_1 X $$
means "For each observation "i" in the dataset, we estimate (that's what the hat on Y means) that this observation's Y value equals  $\beta_1$ multiplied by that observation's X value, plus $\beta_0$."
Obviously, we know that this estimate probably going to be wrong for most observations (that's why it's an estimate). And in any given dataset it is going to be wrong by $\epsilon_i$ for each observation. But if all we want to talk about is our estimate, then this is a fine way to write the model.
