Graphing using Basic shapes and shifting?

Question

How do I draw a rough sketch using basic shapes and shifting of this funciton:

f(x)= |x|-2

Answer 1

Start with a graph of:
f(x) = x

That's a straight line going through the origin with a slope of 1. It should look like a 45 degree angle if you draw it correctly.

Now to transform this to:
f(x) = | x |

You have to think of each value smaller than 0. Rather than -1, for example, you'll have 1. Rather than -2, you'll have 2, etc. This is like "mirroring" the graph that is below the x-axis so that it is above the x-axis.

This results in a "V" shape. A slope of -1 down to the origin, then a slope of 1 from the origin.

Finally:
f(x) = | x | - 2

For every value that results, you now subtract 2. For example, x = 0, gives -2, rather than 0. This is true for every point x, making every point y smaller by 2. It would look like you shifted the whole graph down 2 units.

So it would be "V" like before, but it would go down at a slope of -1, cross the x-axis at -2, and continue to (0, -2). Then it would slope up (m=1) through the x-axis at 2 and continue upward.

You can go to the first link and type 'abs(x) - 2' in the box. The resulting graph is shown as the 2nd link.

Answer 2

Well |x| = x for x ≥ 0
= - x for x < 0

So y = |x| is the line y = x for all x ≥ 0 and the line y = -x for x < 0. This looks like V with the point at (0, 0)
Now subracting 2 from all the y values is going to lower the V on the number plane to the point (0, -2) ie to -2 on the y - axis so it looks something like this:
..............^ y..........
\.............|............./
..\...........|.........../
....\.........|........./
......\.......|......./
------\ ---|.---/--> x
.........\....|..../
...........\..|../
.............V.-2
..............|

(please excuse the attempted graphic but the idea is there)

Algebraically you would describe it as follows:
y = |x| - 2 means
y = x - 2 for x ≥ 0
= - x - 2 for x < 0