Answer:
[tex]y=\left \{ \begin{array}{l}-x^2-4x-1,\ x<-1 \\ \dfrac{1}{2}|x-1|-1,\ x\ge -1\end{array} \right.[/tex]
Step-by-step explanation:
Left part of the graph is the graph of the parabola passing through the points (-2,3), (-3,2) and (-4,-1). If the equation of the parabola is [tex]y=ax^2+bx+c,[/tex] then
[tex]3=a\cdot (-2)^2+b\cdot (-2)+c\Rightarrow 3=4a-2b+c\\ \\2=a\cdot (-3)^2+b\cdot (-3)+c\Rightarrow 2=9a-3b+c\\ \\-1=a\cdot (-4)^2+b\cdot (-4)+c\Rightarrow -1=16a-4b+c[/tex]
Subtract first two equations and last two equations:
[tex]-1=5a-b\\ \\-3=7a-b[/tex]
Suybtract these two equations:
[tex]-2=2a\Rightarrow a=-1[/tex]
So [tex]-1=5\cdot (-1)-b\Rightarrow b=-5+1=-4[/tex]
Substitute into the first equation:
[tex]3=4\cdot (-1)-2\cdot (-4)+c\Rightarrow c=3+4-8=-1[/tex]
The equation of the parabola is [tex]y=-x^2-4x-1[/tex]
The right part of the graph is translated 1 unit to the right and 1 unit down graph of the function [tex]y=\dfrac{1}{2}|x|[/tex], so it has the equation [tex]y=\dfrac{1}{2}|x-1|-1[/tex]
Hence, the piece-wise function is
[tex]y=\left \{ \begin{array}{l}-x^2-4x-1,\ x<-1 \\ \dfrac{1}{2}|x-1|-1,\ x\ge -1\end{array} \right.[/tex]
