Question

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

Answer 1

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