When to report quadratic versus linear relationships I seem to remember from my graduate statistics course that if higher order variables (i.e., X^2, X^3, etc) are significant in a polynomial regression analysis such as our quadratic regression, then the relationship between the DV and IVs is considered to be the highest order variable. 
In other words, when I do a regression in the format of X + X^2, and both the linear (X) and quadratic (X^2) components of the analysis are significant, we report the relationship as quadratic? 
Both the X and X^2 predictors are significant in the model
but X is more significant is it still considered to be a quadratic relationship? Note also that a simple linear regression has a lower R^2 than the quadratic regression.
 A: If this is primarily a linguistic question 'What do I call it?' then I think you use the highest term. So if when you plot it the appearance is almost straight but with a slight curve it is still quadratic.
Some of the other issues about inclusion of terms of various orders have been dealt with extensively on this site, for instance here
A: From my personal experience, I would choose one or another depending on two things. 
First, depending on the application, the error between the regression line and the data that can be accepted may be different. If choosing just linear regression meets your error requirements, why not keeping it simple. It is usually a tradeoff between the precision of fit and the robustness (the ability to applicate it to other sets of same kind of data).
Second, sometimes we have some a priori knowledge on the data. For example, we know that the relationship between y and x is expected to be linear, or that it might be strongly non linear, etc. This may help also to choose between the regression models.
