problemas de dimensionamento em TIKZ

Question

Estou tendo alguns problemas de escala ao desenhar diagramas usando TIKZ. Eu tenho um casco convexo. Desejo desenhar uma parábola passando pelos dois primeiros vértices diferentes de zero do casco convexo (aeb respectivamente). Calculei os coeficientes, de modo que a parábola passe pelos dois primeiros vértices diferentes de zero do casco convexo. Embora eu tenha uniformizado a escala, não consigo obter o resultado desejado. Aqui está MWE

    \documentclass[crop,tikz]{standalone}
    \begin{document}
    
    \begin{tikzpicture}[scale=.35]
    \draw[draw=none,scale=5](0,0)--(-1.1730,0.9848)--(-1.5,0.8660)--(-1.7660,0.6428)--(-1.9397,0.3420)--(-1.9397,-0.3420)--(-1.7660,-0.6428)--(-1.5,-0.8660)--(-1.1730,-0.9848)--(0,0)--cycle;
    \draw[thick,scale=5,color=black]  (0,0)--(-1.1730,0.9848)--(-1.5,0.8660)--(-1.7660,0.6428)--(-1.9397,0.3420); 
    \draw[thick,scale=5]  (-1.9397,-0.3420)--(-1.7660,-0.6428)--(-1.5,-0.8660)--(-1.1730,-0.9848)--(0,0);
    
    
    \draw[scale=5, domain=-2:2, smooth, variable=\y, red]  plot ({1.48720*\y*\y+2.615340}, {\y});
    
    \node[] at (0.75,0.75) {$0$};
    \node[] at (-5.75,5.5) {$a$};
    \node[] at (-7.5,5.0) {$b$};
    
    \draw[thick,->,,>=stealth] (-11.5,0)--(2,0) node[right]{$X$};
    \draw[thick,->,,>=stealth]  (0,-8)--(0,8) node[above]{$Y$};
    
    \end{tikzpicture}
    
    \end{document}

Eu quero algo como:

Answer 1

Tento reproduzir suas imagens:

\documentclass[margin=3mm, tikz]{standalone}

\begin{document}
    \begin{tikzpicture}[
  T/.style = {font=\scriptsize, fill=white, inner sep=1pt},% ticks
dot/.style = {circle, fill, inner sep=1pt},
     scale = 2]
% axis
\draw[thick,-stealth] (-3,0) -- (2,0) node[right] {$X$};
\draw[thick,-stealth] (0,-2) -- (0,2) node[above]{$Y$};
\foreach \i in {-2.5,-2,...,1.5}
    \draw (\i,1mm) -- ++ (0,-2mm) node[T, below] {\i};
\foreach \i in {-1.5,-1,-0.5,0.5,1,1.5}
    \draw (1mm,\i) -- ++ (-2mm,0) node[T,  left] {\i};
% rotated parabola
\draw[domain=0:1.2, variable=\y, 
      red, very thick] (-3,0) -- plot (3*\y*\y-2.5, \y);%- 1.00456
% a and b points on parabola
\node (a) [dot,label=$a$] at (3*0.5*0.5-2.5,0.5) {};
\node (b) [dot,label=$b$] at (3*0.6*0.6-2.5,0.6) {};
% line
\draw   (a) -- ++ (-0.4,-0.3)   node[dot] {} 
            -- ++ (0,-1.2)      node[dot] {} 
            -- ++ (0.4,-0.2)    node[dot] {} 
            -- ++ (0.5,-0.1)    node[dot] {} 
            -- (0,0)            node[dot] {} 
            -- (b);
    \end{tikzpicture}
\end{document}