Exercise 7 Page 159

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. This 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 $y\text{-}$axis. This will transform the graph of the parent function, $y=|x|,$ to the graph of $y=|\text{-}x|.$ Notice that, since the graph of $y=|x|$ is symmetrical about $y\text{-}$axis, this will not change how it looks.

Now, let's do a horizontal translation $4$ units to the right. Then the graph of $y=|\text{-}x|$ becomes $y=|\text{-}(x-4)|.$

Next, we will do a vertical shrink by a factor of $2.$ This transforms the graph of $y=|\text{-}(x-4)|$ to the graph of $y=\frac{1}{2}|\text{-}(x-4)|\iff y=\left|\text{-}\frac{1}{2}(x-4)\right|.$

Finally, we have a vertical translation $1$ unit up. From $y=\left|\text{-}\frac{1}{2}(x-4)\right|$ we get the final graph, $y=\left|\text{-}\frac{1}{2}(x-4)\right|+1.$

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.