Parent function


In mathematics, a parent function is the simplest function of a family of functions that preserves the definition of the entire family. For example, for the family of quadratic functions having the general form
the simplest function is
This is therefore the parent function of the family of quadratic equations.
For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of y = x − 4x + 7 can be obtained from the graph of y = x by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = .
For many trigonometric functions, the parent function is usually a basic sin, cos, or tan. For example, the graph of y = A sin + B cos can be obtained from the graph of y = sin by translating it through an angle α along the positive X axis, then stretching it parallel to the Y axis using a stretch factor R, where R = A + B. This is because A sin + B cos can be written as R sin.
The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x, or, to simplify further, x when n is even and x for odd n. Turning points may be established by differentiation to provide more detail of the graph.