Function transformation rules

Learning Objectives Identify and describe the effect of vertical and horizontal translations, reflections, and dilations from a function's equation. Apply a sequence of transformations to a parent function to sketch its graph accurately. Determine the equation of a transformed function given its graph and the parent function. Analyze how transformations affect key features of a function, including its domain, range, intercepts, and asymptotes. Correctly apply the order of operations for combined transformations, particularly distinguishing between dilations/reflections and translations. Write a new function rule given a set of described transformations. Factor the input of a function to correctly identify horizontal dilations and translations. Ever wondered how video...

TermDefinitionExample Parent FunctionThe simplest, un-transformed version of a function in a family. It serves as the starting point for all transformations.For the family of quadratic functions, the parent function is f(x) = x². For trigonometric functions, it could be f(x) = sin(x) or f(x) = cos(x). TransformationAn operation that alters the size, shape, position, or orientation of a function's graph.Shifting the graph of f(x) = x² two units to the right to get g(x) = (x - 2)². Translation (Shift)A rigid transformation that moves every point on the graph a constant distance horizontally, vertically, or both, without changing its shape or orientation.The graph of f(x) = |x| + 3 is a vertical translation of f(x) = |x| up by 3 units. ReflectionA rigid transformation that flips the gra...

General Transformation Formula g(x) = a f(k(x - d)) + c This formula combines all transformations. 'f' is the parent function. 'a' controls vertical stretch/reflection, 'k' controls horizontal stretch/reflection, 'd' controls horizontal shift, and 'c' controls vertical shift. The standard order of application is dilations/reflections (a and k) first, then translations (d and c). Vertical Transformations y -> ay + c These transformations affect the y-coordinates and follow standard arithmetic. 'a' multiplies the y-values (if |a| > 1, it's a stretch; if 0 < |a| < 1, it's a compression; if a < 0, it's a reflection across the x-axis). 'c' adds to the y-values (if c > 0, shift u...