Centering the x and y axes in the xy plane. And adding legends to function graphs

Question

I am trying to reproduce the picture below on the left with the packages "tikz" and "pgfplots". But as you can see the best I got was the figure below on the right. In my attempt I modified the tex code from here and adapted it as follows below after the pictures.

My questions are as follows.

1. How do I translate the "x" and "y" axes such that these axes intersect perpendicularly at the point (0,0)?

2. How do I position (x,y) coordinates on the graph? How do I position the red circles on the graph?

3. How do I control the size of the axis legends? And the size or width of the axes? How to change it?

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}

\pgfplotsset{compat = newest}

\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            xmin = -6, xmax = +8,
            ymin = -2.0, ymax = +11.0,
            xtick distance = 1.0,
            ytick distance = 1.0,
            grid = both,
            minor tick num = 1,
            major grid style = {lightgray!25},
            minor grid style = {lightgray!25},
            width =  1.0\textwidth,
            height = 1.0\textwidth,
            xlabel = {$x$},
            ylabel = {$y$},
            legend cell align = {left},
        ]
            \addplot[
                domain = -4:3,
                samples = 200,
                smooth,
                blue,
                thick,
            ] %{exp(ln(x+1)/ln(2))};
            {2^(x)+1};
            
            \addplot[
                domain = -4:3,
                samples = 200,
                smooth,
                red,
                dashed
            ] {2^(x)};
            
            %\legend{Plot from expression, Plot from file}
        \end{axis}
    \end{tikzpicture}
\end{document}

Answer 1

1. axis lines=center

2. There are multiple ways of doing that, you could for example use TikZ code such as \node [circle, fill=red, minimum size=5pt, inner sep=0pt, label={[red]below right:$(0,1)$}] at (0,1) {};. However, I think I would rather use the features of pgfplots, specifically nodes near coords. See the code below for an example, ask if anything is unclear.

3. For example legend style={font=\small} to reduce the font size of the legend. The rest of this point you have already answered in your code (the width and height keys)

\documentclass{standalone}

\usepackage{pgfplots}

\pgfplotsset{compat = 1.18} % using an explicit version is recommended by the package author

\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            axis lines=center, % axis lines through origin instead of box
            xmin = -6, xmax = +8,
            ymin = -2.0, ymax = +11.0,
            xtick distance = 1.0,
            ytick distance = 1.0,
            grid = both,
            minor tick num = 1,
            major grid style = {lightgray!25},
            minor grid style = {lightgray!25},
            width =  1.0\textwidth,
            height = 1.0\textwidth,
            xlabel = {$x$},
            ylabel = {$y$},
            legend cell align = {left},
            visualization depends on=x\as\tmpX, % make the x coordinate available for use in nodes near coords below
            visualization depends on=y\as\tmpY, % same for y
            legend style={
               font=\small, % change font size of legend
               at={(1,1)},
               anchor=north east
               }
        ]
            \addplot[
                domain = -4:3,
                samples = 200,
                smooth,
                blue,
                thick,
            ] %{exp(ln(x+1)/ln(2))};
            {2^(x)+1};
            
            \addplot[
                domain = -4:3,
                samples = 200,
                smooth,
                red,
                dashed
            ] {2^(x)};
            
            \addplot [
                blue,
                mark=*,
                only marks,
                samples at={0,1,2,3}, % specify x values to plot function at
                % now add a node next to each plotted point
                % we use the macros defined with visualization depends on here
                % to make the coordinate pair
                nodes near coords={$(\pgfmathprintnumber{\tmpX}, \pgfmathprintnumber{\tmpY})$},
                % finally define were the nodes are placed relative to the plotted points
                nodes near coords align=above left
                ] {2^(x)+1};
            

            % a more manual way of adding dots with coordinate label
            %\node [circle, fill=red, minimum size=5pt, inner sep=0pt, label={[red]below right:$(0,1)$}] at (0,1) {};
            
            \legend{Plot from expression, Plot from file}
        \end{axis}
    \end{tikzpicture}
\end{document}