a We can look at each of the transformations individually and work our way from the inside to the outside.
g(x)=1/3|x-1|-2
The given equation is a transformation of the parent function f(x)=|x|. We will start with the transformation closest to x and end with the farthest from x. First, there is a horizontal translation 1 unit to the right.
|x|⇒|x- 1|
Remember, typically |x-h| is a horizontal translation h units to the right, while |x+h| is translated h units to the left.
Next, we have a vertical shrink by a factor of 3. This transforms the graph of y=|x-1| to the graph of y= 13|(x-1)|.
Finally, we have a vertical translation 2 units down. From y= 13|(x-1)| we get the final graph, y= 13|x-1|-2.
b To graph the function without going through the entire process of transforming the parent function, we can make a table of values. Then, we only need to plot the ordered pairs that we find.
x
1/3|x-1|-2
Simplify
g(x)
-2
1/3|( -2)-1|-2
1/3|-3 |-2
-1
1
1/3|( 1)-1|-2
1/3|0 |-2
-2
4
1/3|( 4)-1|-2
1/3|3 |-2
-1
Now, we can plot these points and connect them to create our graph of g(x).
