To graph the inequality, we should first determine its boundary line. This can be done by replacing the inequality symbol with the equals sign.
&Inequality &&Boundary Line
&y ≤ 2|x-1|-4 &&y = 2|x-1|-4
The graph of the boundary line is the graph of the parent function y=|x| after several transformations. Let's recall these transformations.
Transformations of y=|x|
Vertical Translations
Translation up k units, k>0
y=|x|+ k
Translation down k units, k>0
y=|x|- k
Horizontal Translations
Translation right h units, h>0
y=|x- h|
Translation left h units, h>0
y=|x+ h|
Vertical Stretch or Compression
Vertical stretch, a>1
y= a|x|
Vertical compression, 0< a<1
y= a|x|
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=| bx|
Horizontal compression, b>1
y=| bx|
Reflections
In the x-axis
y=- |x|
In the y-axis
y=|- x|
To graph the boundary line, we will begin by choosing two points on the parent function graph. One of these points should be the vertex. Notice that the inequality is non-strict, so the boundary line will be solid.
We will begin by translating the graph 1 unit to the right. To do that we will add 1 to the x-coordinates of points that form the graph.
Next, we will stretch the graph of y=|x-1| vertically by the factor of 2 by multiplying the y-coordinates of the points that form the graph by 2.
Now we will translate the graph 4 units down by subtracting 4 from the y-coordinates of the points that form the graph.
Finally, we can have the graph of boundary line by removing the unnecessary parts.
To decide which region we should shade, we will test the point (0,0).
