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Maths Exercices

Absolute Value Graphs

In this section, we will study the graphs of functions which contain an absolute value.
When you want to graph a function but are unsure of how that function works, you can always start with a table of values.

Let's graph f(x)=|x| using a table of values:

x
-2
-1
0
1
2
|x|
2
1
0
1
2

Let's plot these points on a graph:

These points are going to be connected with straight lines to get the final graph of f(x)=|x|:

It will be useful to remember that whenever you have an absolute value graph, the general shape will look like a "v" (or in some cases, an upside down "v")


Some transformations can be applied to the graph of the absolute value function. It will be useful to remember that the standard form of the absolute value function is:

y = a|x - h| + k
where
a is vertical stretch
h is horizontal shift
k is vertical shif

Sketch the graph of f(x) = 2|x - 3| -4

For this graph we can start by sketching the basic graph of f(x) = |x| and focus on three important points: (0,0), (-1,1) and (1,1).

Now start by applying the transformations in order:

  1. In this example, a is 2. Multiply all the y - coordinates by 2. The important points
    ((-1,1), (0,0), (1,1)) become (-1,2), (0,0) and (1,2).
    The new graph looks as follows:



  2. In this example, h is positive 3. This means that every point on this graph should be shifted 3 units to the right. This gives you the following graph:

  3. The last transformation to be performed is k, which is -4 in this example. This means that we have to move all of the points DOWN 4 units. This will give us our final graph:



Match the next graph to the correct equation