Compare the functions in the family ga(x)=xn+ax for several fixed values of n.
Graphs are the natural way to begin this investigation, which immediately requires students to translate the algebraic description into a graphical one. This problem also tempts students to form a hasty generalization: adding ax to a power function translates the power function. Further investigation reveals that this is false. There are algebraic manipulations (e.g., completing the square) that demonstrate why the family of quadratic functions ga is made up of translations of x2. The problem is also open-ended enough that students must decide for themselves what aspects to investigate and how to do so.
When students consider the family having n=2, they just think of the functions as translations of each other, but this does not generalize to other values of n.