Library+of+Functions-Target+A-Great+Abs-Guided+Learning

**Target A: Graph Absolute Value Functions and Identify the Domain and Range**

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What is an absolute value function? What does the graph of an absolute value function look like? Take a look at the following function, table, and graph.

As you can see, the graph of the absolute value function is in the shape of a V. The vertex of the graph is the point at which the graph changes direction. In this example, the vertex is located at (0, 0) because that is the point at which the graph changes direction.

The standard equation for absolute value functions is:



The "a" value is essentially the slope of the function, although as we will see later, we cannot define the slope to be exactly the same as the a value, because the slope changes depending on which portion of the graph you are looking at (the different portions are called intervals). The h and k form an ordered pair (h, k) which represents the vertex of the graph of an absolute value function. When graphing an absolute value function, these are the three pieces of information that need to be identified in order to graph your function correctly. **Graphing an Absolute Value Function** When graphing an absolute value function, identify the a, h, k, from the equation. Start by plotting the vertex first, then use the a value to graph the rays that form the absolute value graph. Take a look at the video to see an example.

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media type="youtube" key="-qWv_x1JQH8" height="360" width="640" **Identifying Domain and Range of an Absolute Value Function**  The domain of an absolute value function is always the set of all real numbers. The reason being is that the graphs of the absolute value functions will continue on forever in the left and right directions, eventually spanning all possible x-values. The range of an absolute value function is a little different. The range is determined by the location of the vertex of the graph (specifically the y-value of the vertex). The range of the function will be written as an inequality. If the graph opens up, then the range is the set of all possible numbers that are greater than or equal to the y-value of the vertex. If the graph opens down, then the range is the set of all possible numbers that are less than or equal to the y-value of the vertex. Take a look at the graphs below. Graph A:  Domain: {all real numbers}....the domain of an absolute value function is always the set of all real numbers   Range: {y __>__ -4}....the graph opens up and the y-value of the vertex is -4 Graph B:  Domain: {all real numbers}....the domain of an absolute value function is always the set of all real numbers   Range: {y __<__ 5}....the graph opens down and the y-value of the vertex is 5 Since there is a point on an absolute value graph (vertex) where the graph changes direction, the graph of an absolute value function can either be increasing (going up from left to right) or decreasing (going down from left to right). It can change depending on the interval. An interval is basically of span of the graph which is determined by a set of x-values. Let's take a look at the graph of g(x) = -1/2|x + 1| + 7 below to determine if the function is increasing or decreasing. To determine if the function is increasing or decreasing over the interval -8 __<__ x __<__ -4, we need to look at the portion of the graph that spans from where x has a value of -8 to where x has a value of -4. You can see in the diagram that red dashed lines show this piece of the graph. Next, look at what's happening with the graph as we go from -8 to -4. That portion of the graph is going UP as we move from LEFT to RIGHT. So we can say that the function g(x) = -1/2|x + 1| + 7 is increasing over the interval -8 __<__ x __<__ -4. The second interval given is 3 __<__ x __<__ 9. This means we want to look at the portion of the graph that spans from where x has a value of 3 to where x has a value of 9. This interval is shown by the green dashed lines. As we go from where x is 3 to where x is 9, you can see that the graph of the function is going DOWN as we move from LEFT to RIGHT. Because the graph is going down from 3 to 9, we can say that the function g(x) = -1/2|x + 1| + 7 is decreasing over the interval 3 __<__ x __<__ 9. **Library of Functions-Target A-Great Abs-Quick Check** Given the function h(x) = 2|x + 3| - 6, a) graph the function b) identify the domain and range of the function c) determine if the function is increasing or decreasing over the interval -7 __<__ x __<__ -4 Library of Functions -Target A -Great Abs-Quick Check Solutions
 * Determining if an Absolute Value Function is Increasing or Decreasing**

Library of Functions-Target A Extension - Writing Absolute Value Functions