Parent functions
Most graphs you meet are a few basic shapes moved, stretched or flipped. Which basic shapes are they, and what does each look like?
Where you are 3 inside
What this area is about
Each family of functions has a plainest member, its parent: y = x for lines, y = x^2 for parabolas, \displaystyle y = \sqrt{x} for square root functions, and so on. Shifting, stretching or reflecting a parent’s graph gives other members of its family, so knowing a handful of parent graphs by heart saves plotting each new function from scratch. For each parent, the things to know are its shape, its domain and range, its intercepts and its symmetry.
How these fit together
All three children are definitions, each pairing two parents. The identity and constant functions are y = x, which gives back its input, and y = c, which gives the same output for every input. The squaring and cubing functions are y = x^2, which loses the sign of its input, and y = x^3, which keeps it. The square root and reciprocal functions are \displaystyle y = \sqrt{x}, which accepts no negative input, and \displaystyle y = \frac{1}{x}, which does not accept 0; both graphs show the gap. The absolute value parent, y = |x|, is the absolute value graph.
A path through it
The identity and constant functions first, then squaring and cubing, then the square root and reciprocal functions.
Next: graphing by transforming a parent graph, and the families built on these parents, such as higher degree polynomials, square root functions and transformed reciprocal graphs.