Question

How do you write `y = |6 + 2x| + 1` as a piecewise function?

Answer 1

`y=-(6+2x)+1;x <-3` and
`y=(6+2x)+1;x >-3`

Explanation:

We could say that the absolute equation is shifted 1 units up but in the same x axis. So, We set the expression inside the bar to zero because the absolute means distance from zero but not less than zero. Hence,
`abs (6+2x)=0`
`Or, 6=-2x`
Dividing both sides by `-2`
`6/-2=x`
`:.x=-3`

Hence, the graph will bounce off at `x=-3`
Which divides the graph into two intervals. `(-oo,-3) and (-3,oo)`.

Note that we aren't including `-3` because it will simply turn the expression inside the bar to be zero.
Hence our final piecewise function is.
`y=-(6+2x)+1; if x <-3`
`y=(6+2x)+1; if x>-3`

Note that we are excluding the vertical shift of `1` units because it doesn't afflict the equation.