Transformations of functions

Learning Objectives Identify and describe the effects of vertical and horizontal translations, reflections, and dilations on a parent function. Apply transformations to a given function f(x) in the correct order to produce the graph of y = a * f(k(x - d)) + c. Determine the equation of a transformed function given its graph and the parent function. Use mapping notation to determine the new coordinates of a point after a series of transformations. Analyze and state the impact of transformations on the domain, range, and other key features (e.g., period, amplitude, asymptotes) of a function. Apply transformations to various families of functions, including polynomial, radical, rational, trigonometric, exponential, and logarithmic functions. How can one simple equation like y =...

TermDefinitionExample Parent FunctionThe simplest, most basic function in a family, from which more complex functions are derived through transformations.For the family of parabolas, the parent function is f(x) = x². For trigonometric functions, it could be f(x) = sin(x) or f(x) = cos(x). TranslationA rigid transformation that shifts a graph horizontally (left or right) or vertically (up or down) without changing its shape, size, or orientation.The graph of g(x) = x² + 3 is the graph of f(x) = x² translated 3 units up. ReflectionA transformation that flips a graph across a line, such as the x-axis or y-axis, creating a mirror image.The graph of g(x) = -√x is a reflection of the parent function f(x) = √x across the x-axis. DilationA non-rigid transformation that stretches or compresses a g...

General Transformation Formula y = a \cdot f(k(x - d)) + c This is the master formula for all transformations. 'f' represents the parent function. The parameters a, k, d, and c dictate the specific transformations. It is crucial to apply them in the correct order: dilations and reflections (a and k) first, followed by translations (d and c). Vertical Transformations (a and c) y = a \cdot f(x) + c These transformations affect the y-values. 'a' controls vertical stretch/compression and reflection across the x-axis. If |a| > 1, it's a stretch. If 0 < |a| < 1, it's a compression. If a < 0, it's a reflection. 'c' controls the vertical shift (up for c > 0, down for c < 0). Horizontal Transformations (k and d) y =...