How to Plot a Parabolic-like Curve Passing Through Specific Points in TikZ or tkz-fct?

Question

I am trying to plot a smooth, parabolic-like curve in LaTeX using TikZ or tkz-fct, and I need the curve to pass through the following specific points: (-2,0), (-1.5,-1), (-1,0),, (-0.5,0.5), (0,0), (0.5,-1), and (1,0) Horizontal arrows at (-1.5,-1) (-0.5,0.5) (0.5,-1) to looks like this one

I attempted to use a polynomial approximation, but the results were not smooth enough or parabolic in shape. The function I tried did not capture the correct curve through these points.

Here’s a simplified version of my attempts:

With pgfplots

\documentclass[tikz,border=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}

\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            axis x line=middle,
            axis y line=middle,
            grid=both,
            xmin=-3, xmax=3,
            ymin=-2, ymax=2,
            xtick={-2,-1,0,1,2},
            ytick={-1,0,1},
            width=10cm,
            height=6cm,
            domain=-2.5:2.5,
            samples=200,
            smooth,
            thick,
            every axis plot/.append style={thick},
            enlargelimits
            ]
            
            % Smooth curve with parabolic behavior passing through the points
            \addplot[black,thick,smooth] coordinates {
                (-2,0) (-1.5,-1) (-1,0) (-0.5,0.5) (0,0) (0.5,-1) (1,0)
            };
            % Horizontal arrows at the requested points
            % Arrow at (-1.5, -1)
            \draw[<->,thick] (axis cs:-1.8,-1) -- (axis cs:-1.2,-1);
            % Arrow at (-0.5, 0.5)
            \draw[<->,thick] (axis cs:-0.8,0.5) -- (axis cs:-0.2,0.5);
            % Arrow at (0.5, -1)
            \draw[<->,thick] (axis cs:0.2,-1) -- (axis cs:0.8,-1);
            
        \end{axis}
    \end{tikzpicture}
\end{document}

With tkz-fct

\documentclass{standalone}
\usepackage{tkz-fct}

\begin{document}
\begin{center}
\begin{tikzpicture}[gridded]
    \tkzInit[xmin=-3,xmax=3,ymin=-2,ymax=2]
    \tkzAxeX \tkzAxeY
    \tkzGrid

    % Plotting the approximated smooth curve passing through the points
    \tkzFct[domain=-2:1,samples=1000,draw=black,thick]{0.84375*x**6 - 0.9375*x**5 - 3.28125*x**4 + 3.75*x**3 + 2.84375*x**2 - 1.375*x}

    % Adding filled circles at specific points
    \draw[fill=black] (-2,0) circle (2pt);   % Point (-2, 0)
    \draw[fill=black] (-1,0) circle (2pt);   % Point (-1, 0)
    \draw[fill=black] (0,0) circle (2pt);    % Point (0, 0)
    \draw[fill=black] (1,0) circle (2pt);    % Point (1, 0)

    % Adding open circle at (0.5, -1)
    \draw[fill=white,draw=black] (0.5,-1) circle (2pt);

    % Horizontal arrows at specific points
    \draw[<->,thick] (-1.8,-1) -- (-1.2,-1); % Arrow at (-1.5, -1)
    \draw[<->,thick] (-0.8,0.5) -- (-0.2,0.5); % Arrow at (-0.5, 0.5)
    \draw[<->,thick] (0.2,-1) -- (0.8,-1); % Arrow at (0.5, -1)

\end{tikzpicture}
\end{center}
\end{document}

Unfortunately, this function doesn't fully capture the smooth parabolic-like shape I'm looking for. How can I create a parabolic-like curve that passes through the points above while keeping it smooth, preferably using tkz-fct or TikZ?

Any suggestions would be greatly appreciated!

Answer 1

Using the approach from my comment, I got the following solution. While it passes through all the points, it "should" have a slope of zero at two of them.

It would certainly be easier just to move the points to match the polynomial.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.17}

\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            axis x line=middle,
            axis y line=middle,
            grid=both,
            xmin=-3, xmax=3,
            ymin=-2, ymax=2,
            xtick={-2,-1,0,1,2},
            ytick={-1,0,1},
            width=10cm,
            height=6cm,
            domain=-2.5:2.5,
            samples=200,
            smooth,
            thick,
            every axis plot/.append style={thick},
            enlargelimits
            ]
            
            % Smooth curve with parabolic behavior passing through the points
            \addplot[black,thick] {(x+2)*(x+1)*x*(x-1)*(0.17746*(x+0.5)*(x+0.5)+0.88888)};
            \addplot[only marks] coordinates  {(-2,0) (-1.5,-1) (-1,0) (-0.5,0.5) (0,0) (0.5,-1) (1,0)};

            % Horizontal arrows at the requested points
            % Arrow at (-1.5, -1)
            \draw[<->,thick] (axis cs:-1.8,-1) -- (axis cs:-1.2,-1);
            % Arrow at (-0.5, 0.5)
            \draw[<->,thick] (axis cs:-0.8,0.5) -- (axis cs:-0.2,0.5);
            % Arrow at (0.5, -1)
            \draw[<->,thick] (axis cs:0.2,-1) -- (axis cs:0.8,-1);
            
        \end{axis}
    \end{tikzpicture}
\end{document}