mjhjmtu

How can I improve this curve? Why do the arrows ruin the left side but keep the right side smooth? All suggestions welcome.

documentclassarticle
usepackagetikz
begindocument
begintikzpicture[scale=.6,cap=round]
 tikzsetaxes/.style=
 % The graphic
 beginscope[style=axes]
 draw[->] (-5,0) -- (5,0) node[below] $x$;
 draw[->] (0,-3)-- (0,5) node[left] $y$;
 %draw [gray!50] (-5,3) -- (-2.5,1) -- (2.5,4) -- (5,-1);
 draw [red,thick,<->] plot [smooth,samples=200, tension=1] coordinates 
 (-5,4) (-2.5,1) (2.5,4) (5,-1);
 endscope
 endtikzpicture
 enddocument

I do not precisely understand what you mean by "Why do the arrows ruin the left side but keep the right side smooth? ". However, this is just to say that there are many possible ways to draw smooth curves with TikZ. A rather powerful way is to employ the hobby library.

documentclassarticle
usepackagetikz
usetikzlibraryhobby
begindocument
begintikzpicture[scale=.6,cap=round,
tangent/.style=%
in angle=(180+#1) ,
Hobby finish ,
designated Hobby path=next , out angle=#1,
]
 tikzsetaxes/.style=
 % The graphic
beginscope[style=axes]
 draw[->] (-5,0) -- (5,0) node[below] $x$;
 draw[->] (0,-3)-- (0,5) node[left] $y$;
 %draw [gray!50] (-5,3) -- (-2.5,1) -- (2.5,4) -- (5,-1);
 draw [red,thick,<->,use Hobby shortcut] 
 ([tangent=-80]-5,4) .. (-2.5,1) .. (2.5,4) .. ([tangent=-80]5,-1);
endscope
endtikzpicture
enddocument

Does this come closer to what you want to achieve?

ADDENDUM: You are asking about the distortion you get when you add or remove an arrow. The effect you are referring to is clearly visible as the difference between the red and blue curves in

documentclassarticle
usepackagetikz
%usetikzlibrarybending
begindocument
begintikzpicture[scale=.6,cap=round]
 tikzsetaxes/.style=
 % The graphic
 beginscope[style=axes]
 draw[->] (-5,0) -- (5,0) node[below] $x$;
 draw[->] (0,-3)-- (0,5) node[left] $y$;
 %draw [gray!50] (-5,3) -- (-2.5,1) -- (2.5,4) -- (5,-1);
 draw [red,thick,<->] plot [smooth,tension=1] coordinates 
 (-5,4) (-2.5,1) (2.5,4) (5,-1);
 draw [blue,thick,-] plot [smooth,tension=1] coordinates 
 (-5,4) (-2.5,1) (2.5,4) (5,-1);
 endscope
endtikzpicture
enddocument

This effect is due to the fact that a (standard) arrow inserts a stretch of straight (as opposed to curved) path. You can simply get rid of it by activating usetikzlibrarybending, i.e. by removing the % in front of it.

Of course, if you have longer arrows, you want to make sure that you load arrows.meta as well and pass bend to the options of the arrow, e.g. `

 draw [red,thick,Latex[bend,length=5pt]-Latex[bend,length=5pt]] plot [smooth,tension=1] coordinates 
 (-5,4) (-2.5,1) (2.5,4) (5,-1);