# Graphs of Absolute Value Equations

Graphing Composite Absolute Value Functions

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• Let's start with a simple one $y = | x + 2 |$

If $x > - 2$, $x + 2$ is positive, so $y = | x + 2 | = x + 2$
If $x < - 2$, $x + 2$ is negative, but will be 'turned around' by the abslote signs, so in this domain $y = | x + 2 | = - x - 2$

These two semi-graphs meet at $\left(- 2 , 0\right)$
graph{|x+2| [-10.5, 9.5, -1.08, 8.915]}

• The shape of a graph with an absolute value would look something similar to a "V".

graph{abs(x) [-10, 10, -5, 5]}

This graph has a slope of $1$ on the right of the $y$=axis and $- 1$ on the left of the $y$-axis

• $y = a \left\mid x - h \right\mid + k$ has vertex $\left(h , k\right)$

$y = 3 \left\mid x - 1 \right\mid + 5$ has vertex $\left(1 , 5\right)$

• A table of values is simply a pairing between an input and an output. Absolute value equations can range from very complex to relatively simple -- there are also many theorems and postulates that dictate the behavior of absolute value equations in various circumstances.

In its purest form, absolute value makes all numbers positive. For instance, $y = | x |$ would result in the positive form of $x$ being the output, whether $x$ is positive or negative. For instance, if $x$ is $- 5$, it would be $5$ when it comes out of that equation. Create a table of values like you normally would, but compute values enclosed in absolute value markers as positive.

Another common form of absolute value equations can be found in a form similar to $| a - 3 | = 5$

Thus:

$| a - 3 | = 5$ would have $a$ values $- 2 < x < 8$

From there, you can create a table from the values of $a$.

If you can update your question with more detail, I may be able to answer it better.

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