Be sure to think about the horizontal transformations before any vertical transformations. Work from the inside to the outside. Then make a table of values to find points on the graph.
Translations:
A horizontal translation 1 unit to the left.
A vertical stretch by a factor of 3.
A reflection in the x-axis.
A vertical translation 4 units up. Graph:
Practice makes perfect
First we will go through all the transformations from the graph of f(x)=|x| to the graph of given function. After that we will make a table of values to graph this function.
Transformations
The given equation is a transformation of the parent function f(x)=|x|. We can define each transformation from the inside to the outside. We will start with the transformation closest to x and end with the farthest from x. Let's review our function.
v(x)=-3|x+1|+4
First, there is a horizontal translation 1 units to the left, |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.
Then, we have a vertical stretch by a factor of 3, |x+1|⇒ 3|x+1|.
Next, we have a reflection in the x-axis, 3|x+1|⇒ -3|x+1|.
Finally, we have a vertical translation 4 units up, -3|x+1| ⇒-3|x+1|+4.
Graphing v(x)
To graph the function without going through the entire process of transforming the parent function, we could instead make a table of values. Then we only need to plot the ordered pairs that we find.
x
-3|x+1|+4
Simplify
v(x)
-3
-3| -3+1|+4
-3|-2|+4
-2
-2
-3| -2+1|+4
-3|-1|+4
1
-1
-3| -1+1|+4
-3|0|+4
4
0
-3| 0+1|+4
-3|1|+4
1
1
-3| 1+1|+4
-3|2|+4
-2
Now we can plot these points and connect them to create our graph of v(x).
