Describe function transformations

Learning Objectives Identify the four basic types of transformations: translations, reflections, and dilations (stretches/compressions). Describe the sequence of transformations applied to a parent function given the equation of the transformed function. Write the equation of a transformed function given a description of the transformations and the parent function. Correctly apply the order of operations for transformations (dilations and reflections before translations). Distinguish between horizontal and vertical transformations based on the location of the parameters in the function's equation. Analyze and describe transformations for a variety of parent functions, including polynomial, radical, rational, trigonometric, and logarithmic functions. How does a video gam...

TermDefinitionExample Parent FunctionThe simplest form of a function in a family, which is then transformed to create more complex versions.For the family of parabolas, the parent function is f(x) = x^2. For logarithmic functions, it is f(x) = ln(x) or f(x) = log(x). TransformationAn operation that changes the graph of a function by altering its position, shape, or orientation.Shifting the graph of y = sin(x) up by 2 units is a transformation. Translation (Shift)A transformation that moves every point on a graph the same distance in the same direction, either horizontally or vertically.The graph of g(x) = (x - 3)^2 is a horizontal translation of f(x) = x^2 three units to the right. ReflectionA transformation that flips a graph across a line, such as the x-axis or y-axis.The graph of g(x)...

General Transformation Formula g(x) = a \cdot f(b(x - h)) + k This formula combines all transformations. 'a' and 'k' affect the graph vertically (outside the function), while 'b' and 'h' affect it horizontally (inside the function). The recommended order of application is: 1. Horizontal shifts (h), 2. Horizontal stretches/compressions and reflections (b), 3. Vertical stretches/compressions and reflections (a), 4. Vertical shifts (k). However, it is often more reliable to apply stretches/reflections first, then translations. Vertical Transformations (Outside) y = a \cdot f(x) + k Operations outside f(x) affect the y-values. 'a' causes a vertical stretch/compression and/or reflection across the x-axis. 'k' causes a ve...