Explaining The Shape Of An Absolute Value Graph

Question

I don't understand why the shape of the graph would be like this, for the give equation. Why wouldn't the shape resemble that of a normal quadratic graph $(abs(x)+3)^2$?

Answer 1

It does resemble two portions of a normal quadratic graph, glued together. Just check the values at integers: $9, 16, 25, \ldots$. However, the apexes of the parabolas are in the cut-off parts. To better see the relation, you may want to graph $(x+3)^2$ and $(-x+3)^2$ concurrently.

Answer 2

$y=(|x|^2+3)^2= x^2 +6|x| +9.$

1) $x \ge 0: $

$y=x^2 +6x +9 =(x+3)^2$

2) $x \lt 0$:

$y = x^2-6x +9=(x-3)^2$

1) $y= (x+3)^2$, is a parabola with vertex at $(-3,0)$.

For $x\ge 0$ you get the right hand branch of your drawing.

2) $y= (x-3)^2$, is a parabola with vertex at $(3,0)$.

For $x\lt 0$ you get the left hand branch of your drawing.