Transformations of absolute value functions

Learning Objectives Identify the parameters a, h, and k in the function g(x) = a|x - h| + k. Describe the effect of each parameter (a, h, k) on the graph of the parent function f(x) = |x|. Graph a transformed absolute value function using its vertex and the value of 'a'. Write the equation of a transformed absolute value function given its graph. Determine the vertex, axis of symmetry, domain, and range of a transformed absolute value function. Compare the properties of a transformed function to the parent function f(x) = |x|. Have you ever noticed how a bouncing ball's path can look like a 'V' shape? 🏀 We can describe that path perfectly using math! In this tutorial, you will learn how to take the basic V-shaped graph of y = |x| and move it, flip...

TermDefinitionExample Parent FunctionThe simplest form of a function in a family. For absolute value functions, the parent function is f(x) = |x|.The graph of f(x) = |x| is a V-shape with its vertex at the origin (0, 0) and sides with slopes of 1 and -1. TransformationA change made to a function's graph, affecting its position, shape, or orientation.Shifting the graph of y = |x| two units to the right to get y = |x - 2| is a transformation. TranslationA transformation that 'slides' a graph horizontally (left or right) or vertically (up or down) without changing its shape or orientation.The graph of y = |x| + 3 is a vertical translation of the parent function 3 units up. DilationA transformation that stretches or compresses a graph, making it narrower or wider. It is control...

The Transformation Formula g(x) = a|x - h| + k This is the general form for a transformed absolute value function, where 'a', 'h', and 'k' are parameters that control the transformations applied to the parent function f(x) = |x|. Parameter 'a': Dilation and Reflection If |a| > 1, the graph is vertically stretched (narrower). If 0 < |a| < 1, the graph is vertically compressed (wider). If a < 0, the graph is reflected across the x-axis. The 'a' value multiplies the output of the absolute value. It controls the steepness of the V-shape and whether it opens up or down. Parameters 'h' and 'k': Translations The vertex of the graph is at the point (h, k). 'h' controls the horizontal sh...