1. 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

2. 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

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

4. 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

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

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

7. 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)

8. 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)