Turning points may be established by differentiation to provide more detail of the graph. 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 n, or, to simplify further, x 2 when n is even and x 3 for odd n. This is because A sin( x) + B cos( x) can be written as R sin( x−α) (see List of trigonometric identities). All that does is shift the vertex of a parabola to a point (h,k) and changes the speed at which the parabola curves by a factor of a ( if a is negative, reflect across x axis, if a0 < a < 1, then the parabola will be wider than the parent function by a factor of a, if a 1, the parabola will be the same shape as the parent function but translated. For example, the graph of y = A sin( x) + B cos( x) can be obtained from the graph of y = sin( x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄ B), then stretching it parallel to the Y axis using a stretch factor R, where R 2 = A 2 + B 2. Quadratic functions together can be called a family, and this particular function the parent, because this is the most basic quadratic function (i.e. This is because the equation can also be written as y − 3 = ( x − 2) 2.įor many trigonometric functions, the parent function is usually a basic sin( x), cos( x), or tan( x). For example, the graph of y = x 2 − 4 x + 7 can be obtained from the graph of y = x 2 by translating +2 units along the X axis and +3 units along Y axis. This is therefore the parent function of the family of quadratic equations.įor 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, for the family of quadratic functions having the general form In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation.
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