Question

What are the vertex and range of y = |x + 2| − 6?

(Please explain how to do this! This is a practice question to study)

Answer 1

Check the 1st picture below, that's just a template for function transformations.

Now, about the 2nd picture

well, the function y = |x + 2| - 6, is really just the same as y = |x| but transformed some, with a value of C = +2 and D = -6. Based on that transformation template, we can look at the derived function like this

`y=|x+2|-6\implies y = \stackrel{A}{1}|\stackrel{B}{1}x\stackrel{C}{+2}|\stackrel{D}{-6} \\\\\\ \begin{cases} B=1\\ C=+2 \end{cases}\implies \cfrac{2}{1}\implies 2\textit{ units shifted to the left} \\\\\\ D=-6\implies \hspace{5em} 6\textit{ units translated downwards}`

so as you can see in the 2nd picture, the parent |x| lands with its vertex down below, its domain is pretty much the same since the function keeps on going to ±infinity horizontally, now, the range changed since vertically it used to be down at 0,0 now it's at -2,-6, so the range will be -6 and anything above that, and we can write that in interval notation as

Domain (-∞ , +∞)

Range [-6 , +∞)