The transformation y=|x|+k vertically shifts the graph of y=|x| by k units. We can arbitrarily choose values for k to determine the direction of the translation.
If k>0, the graph of y=|x|+k is a translation up k units of the graph of y=|x|. For example, y=|x| + 2 is a translation up 2 units of the graph of y=|x|.
If k<0, the graph of y=|x|+k is a translation down k units of the graph of y=|x|. For example, y=|x| - 2 is a translation down 2 units of the graph of y=|x|.
Let's graph the examples we have stated along with y=|x|.
b
The transformation y=|x-h| horizontally shifts the graph of y=|x| by h units. We can arbitrarily choose values for h to determine the direction of the translation.
If h>0, the graph of y=|x-h| is a translation right h units of the graph of y=|x|. For example, y=|x - 3| is a translation right 3 units of the graph of y=|x|.
If h<0, the graph of y=|x-h| is a translation left h units of the graph of y=|x|. For example, y=|x + 3| is a translation left 3 units of the graph of y=|x|.
Let's graph the examples we have stated along with y=|x|.
c
The transformation y=a|x| vertically stretches y=|x| by a factor of a. We can arbitrarily choose values for a to determine how the graph changes.
If a>0, the graph of y=a|x| is a vertical stretch of the graph of y=|x| by a factor of a. For example, the graph of y= 2|x| is a vertical stretch of the graph of y=|x| by a factor of 2.
If a<0, the graph of y=a|x| is a vertical stretch of the graph of y=|x| by a factor of |a|, followed by a reflection in the x-axis. For example, the graph of y= - 2|x| is a vertical stretch of the graph of y=|x| by a factor of 2, followed by a reflection in the x-axis.
Let's graph the examples we have stated along with y=|x| to show both transformations.
d
The transformation y=|ax| horizontally shrinks the graph of y=|x| by a factor of |a|. Since the absolute valueremoves the negative sign, it does not matter whether a is a positive or a negative number. Let's show this by graphing y=|x|, y=|3x|, and y=|- 3x| on the same coordinate plane.
Note that since the absolute value removes the negative sign, y=| 3x| coincides with y=| -3x|.
