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

1 Answer
Aug 10, 2015

Answer:

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