GOAL: To understand and graph absolute value equations
DEFINITION: The absolute value of a number can be defined as:
In order to graph an absolute value equation, we need to know what the graph will look like.
Let's look at the graph of y=abs.value(x)
This graph is v-shaped, and so will any other absolute value graph.
This graph opens upwards, but not all graphs will open upwards.
For example, let's look at the graph of y=-(abs.(x))
This graph actually opens downwards.
In fact, any graph that is positive will open up, while ones that are negative will open down.
The vertex of an absolute value graph will either be the highest or lowest point on the graph, depending on which way it opens.
The vertex of y=abs(x) is (0,0) and is the lowest point on the graph.
The vertex of y=-abs(x) is (0,0) and is the highest point on the graph.
Will all of the vertices be (0,0)?
The answer to this question is no.
Let's look at the graph of y=abs(x+1):
The vertex of this graph is (-1,0).
Try graphing y=abs(x+n) and animating the graph to explore what happens to the vertex.
You can see that adding or subtracting within the absolute value sign translates the graph horizontally.
What will happen if we add or subtract outside of the absolute value sign?
Let's look at the graph of y=abs(x)+1:
Here, the vertex is now (0,1).
Try graphing y=abs(x)+n and animating the graph to explore what happens to the vertex.
1. All graphs are v-shaped
2. Graph will open up if positive, open down if negative
3. Vertex will be shifted horizontally if adding or subtracting within absolute value sign
4. Vertex will be shifted vertically if adding or subtracting outside of the absolute value sign
EXTENSION:
What will happen to the graph if multiplying within the absolute value sign?
What will happen to the graph if multiplying outside the absolute value sign?