Question

How do you write `abs(x^2 +x -12)` as a piecewise function?

Answer 1

Use the definition:
`|A|={(A;A>=0),(-A;A<0):}`
Find the points where the quadratic is zero to simplify the restrictions and add pieces as needed.

Explanation:

Given `y = |x^2 +x -12|`

Use the definition, `|A|={(A;A>=0),(-A;A<0):}`:

`y = {(x^2 +x -12;x^2 +x -12>=0),(-(x^2 +x -12);x^2 +x -12<0):}`

Find the x values for `x^2 +x -12=0`:

Factor:

`(x-3)(x+4)=0`

`x = -4 and x = 3`

This means that `x^2 +x -12 >=0` for `x <= -4` and `x>=3`

Modify the restriction for the first piece to be `x <=-4` and add a third peace with the restriction `x>=3`:

`y = {(x^2 +x -12;x<=-4),(-(x^2 +x -12);x^2 +x -12 < 0),(x^2 +x -12;x>=3):}`

Modify the restriction for the middle piece to be `-4 < x < 3` and distribute the -1

`y = {(x^2 +x -12;x<=-4),(-x^2 -x +12;-4 < x < 3),(x^2 +x -12;x>=3):}`

Here is a graph of `y = |x^2 +x -12|`

Here is a graph of the piece-wise function

`y = {(x^2 +x -12;x<=-4),(-x^2 -x +12;-4 < x < 3),(x^2 +x -12;x>=3):}`

with each piece in a different color: