g(x)=|- 12x+2|+1
This is a transformation of the parent function y=|x|. To see how each of the parameters is affecting the parent function, it will be helpful to rewrite it first.
g(x)=|- 12x+2|+1
⇕
g(x)=|- 12 ( x-4)|+1This way we can look at each of the transformations individually. Notice that there is no general recipe for how to obtain a function from its parent function by transformations. However, there are a couple good things to remember.
Usually, we want to look at horizontal transformations before any vertical transformations.
Begin with the reflections.
First, we can do a reflection in the y-axis. This will transform the graph of the parent function, y=|x|, to the graph of y=|- x|.
Notice that, since the graph of y=|x| is symmetrical about y-axis, this will not change how it looks.
Now, let's do a horizontal translation 4 units to the right. Then the graph of y=|- x| becomes y=|-(x-4)|.
Next, we will do a vertical shrink by a factor of 2. This transforms the graph of y=|-(x-4)| to the graph of y= 12|-(x-4)| ⇔ y=|- 12(x-4)|.
Finally, we have a vertical translation 1 unit up. From y=|- 12(x-4)| we get the final graph, y=|- 12(x-4)|+1.
Notice that there are many ways to obtain the graph of our function transforming the graph of its parent function. Here we considered only one of the many possibilities.
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
|-1/2x+2|+1
Simplify
g(x)
-4
|-1/2( -4)+2|+1
|4|+1
5
-2
|-1/2( -2)+2|+1
|3|+1
4
0
|-1/2( 0)+2|+1
|2|+1
3
2
|-1/2( 2)+2|+1
|1|+1
2
4
|-1/2( 4)+2|+1
|0|+1
1
6
|-1/2( 6)+2|+1
|-1|+1
2
Now we can plot these points and connect them to create our graph of g(x).
