# How do you graph f(x) =abs(x+4)-3?

Aug 10, 2015

graph{|x+4|-3 [-10, 10, -5, 5]}

#### Explanation:

Start from the definition of an absolute value function.
It looks like this:
for any non-negative real number its absolute value is itself;
for any negative real number its absolute value is its opposite.
So, for example, $| 3.14 | = 3.14$, $| 0 | = 0$, $| - 2.71 | = 2.71$ etc.

Let's draw a simple graph of a function $y = | x |$.

For non-negative $x$ it coincides with a function $y = x$, that is, it's a straight line at angle ${45}^{o}$ to X-axis, measured from the positive direction of the X-axis counterclockwise.

For negative $x$ it coincides with a function $y = - x$, that is, it's a straight line at angle ${135}^{o}$ to X-axis, measured from the positive direction of the X-axis counterclockwise or, which is the same, at angle ${45}^{o}$ to X-axis, measured from the negative direction of the X-axis clockwise.

Combining both parts of a graph, we obtain the following graph for function $y = | x |$:

graph{|x| [-10, 10, -5, 5]}

Next step is to transform this graph to $y = | x + 4 |$.
As can be easily observed, the original function $y = | x |$ takes the same values ($y$ values) as the new function $y = | x + 4 |$ for arguments $x$ smaller by $4$. Therefore, the graph of $y = | x + 4 |$ is shifted by $4$ to the left from the original $y = | x |$ and looks like this:

graph{|x+4| [-10, 10, -5, 5]}

The last step is to transform graph $y = | x + 4 |$ to $y = | x + 4 | - 3$.
As can be easily observed, the new function $y = | x + 4 | - 3$ takes the values ($y$ values) smaller by $3$ than the old function $y = | x + 4 |$ for the same arguments $x$. Therefore, the graph of $y = | x + 4 | - 3$ is shifted by $3$ down from the graph of function $y = | x + 4 |$ and looks like this:

graph{|x+4|-3 [-10, 10, -5, 5]}

We recommend to watch the series of lectures about graphs on Unizor by following the menu Algebra - Graphs