An excellent way to remember a parent function is to associate the function to it's graph. The visual aspect plays an important role in helping someone to memorise something.
For instance, for the constant function f(x)=a , the graph is a line parallel to x axis.
If a = 3, the graph of the function f(x)=3 is the red line, parallel to x axis, that intercepts y axis at y=3.
For a linear function, f(x) = ax + b, the graph is a line that is no longer parallel to x axis.
For instance, if f(x) = x + 3, the graph is the red line that intercepts x axis at the point (-3,0) and y axis at (0 , 3).
If the parent function is a quadratic, the graph will be a upward or downward concave parabola, that will intercept the x axis in two distinct points, one point or no point, depending on the nature of the roots of quadratic.
For instance, a quadratic that has two roots,it will look like:
The parabola that has two equal roots it will look like:
You notice that the values of the equal roots gives the location of the vertex, that is tangent to x axis. Of course, there is a downward concave version, also.
You can also keep in mind that the logarithmic function is the inverse function of exponential function and the graph of logarithmic function can be found if we'll mirror the graph of exponential function, with respect to the 1st bisectrix.
Therefore, keep in mind that the visual aspect helps you to remember much more easier the family of parent functions.
Create flashcards with the function written out on one side and the graph drawn out by the other. Create two sets of this and you can play a matching game with them. The act of making the cards will aid in memorization and can serve as a quick quizzing method.