Advanced Algebra Notes Section 2.7: Use Absolute Value Functions and Transformations In this section we will be dealing with absolute value functions.   An absolute value function is defined by f(x) = |x|. This function is called the _______________. The graph of f(x) = |x|is a _________________ figure and is symmetric about the y-axis. So for every point (x, y) on the graph, the point (-x, y) must also be on the graph. The highest point or lowest point on the graph of an absolute value function is called the _________. The vertex of the parent function f(x) = |x| is (0, 0). We can derive new absolute value functions from the parent function by using _________________. parent function V-shaped figure vertex transformations

Transformations change a graphs size, shape, position, and orientation.   A _____________ is a transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape or orientation. The graph , is a translation that shifts the graph _________horizontally and _________ vertically from the origin. The vertex of the graph is _________. The h value is always the ______________ of what you see in the original problem. The tells you two things: If a is positive the graph opens _____ and if negative opens ________. translation h units k units (h, k) opposite sign up down The |a| tells you how wide or narrow the graph is. If |a|= 1 then the graph is the______________, if |a|< 1 then the graph will be _______, and if |a| > 1 the graph will be ___________. base shape wider narrower

Example 1: Graph Vertex : (1, 3) a = 1 , Up |a| = 1 , base shape Line of Symmetry: x = h x = 1

Example 2: Graph the following. A. B. The a value also acts like the slope when getting from one point to another. , up, wider V (0, 0) a = -2 , down , narrower Line of Symmetry: x = 0 V (0, 0) Line of Symmetry: x = 0

We can also have graphs with multiple transformations.   Example 3: Graph , up , wider V(-3, -2) Line of Symmetry: x = -3

Example 4: Write the equation of the absolute value graph pictured.

We can also perform transformations on the graph of any function f in the same way as for absolute value graphs.   The graph of can be obtained from the graph of any function y=f(x) by performing the following steps. Steps: 1. Multiply the a value by the y-coordinate of the given points. 2. Translate the graph from step 1 horizontally by shifting the x-coordinates h units rt./lt. and the y-coordinates vertically k units up/down. Example 5: The graph of a function y = f(x) is shown. Sketch the graph of the given functions .       A) Pts. Of Given Graph (-3, -1) (0, 0) (3, -2) (-3, -3) (0, 0) (3, -6) (-3, 1) (0, 0) (3, 2)

B) (-3, -6) (0, 0) (3, -12) A is not negative so can’t do this step. Shift x 1 unit rt. & y 3 units up. (-2, -3) (1, 3) (4, -9)