Reflections of functions

Learning Objectives Identify the type of reflection (x-axis or y-axis) from a function's equation. Graph a function and its reflection across the x-axis and y-axis. Determine the equation of a reflected function given the original function and the axis of reflection. Analyze the effect of reflections on key features of a function, such as domain, range, intercepts, and turning points. Combine reflections with other transformations like translations and stretches. Use the concept of reflections to algebraically test for even and odd function symmetry. Ever wonder how a kaleidoscope creates its beautiful, symmetrical patterns? 💠 It's all about the mathematics of reflections! This tutorial explores reflections, a fundamental type of function transformation. You will...

TermDefinitionExample ReflectionA transformation that creates a mirror image of a graph across a specific line, known as the axis of reflection.Reflecting the point (3, 4) across the x-axis results in the image point (3, -4). Axis of ReflectionThe line over which a function's graph is flipped. In this context, it is typically the x-axis (the line y=0) or the y-axis (the line x=0).For the transformation y = -f(x), the axis of reflection is the x-axis. Image PointA point on the transformed (reflected) graph that corresponds to an original point on the parent function.If (2, 8) is on the graph of f(x) = x³, then (-2, 8) is the image point on the graph of y = f(-x) after a y-axis reflection. Invariant PointA point on a graph that remains in the same position after a transformation is app...

Reflection Across the x-axis y = -f(x) To reflect a function's graph across the x-axis, multiply the entire function's output by -1. The mapping for any point (x, y) on the original graph is (x, -y) on the reflected graph. Reflection Across the y-axis y = f(-x) To reflect a function's graph across the y-axis, replace every instance of 'x' in the function's formula with '-x'. The mapping for any point (x, y) on the original graph is (-x, y) on the reflected graph.