Why (\x)^2 and \x^2 produce different results in Tikz plot? [duplicate]

Question

(\x)^2 will plot correct parabola, while \x^2 will have negative x branch upside down. Why () are required for this formula to work?

\documentclass[10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{shapes.misc}

\begin{document}
\begin{figure}[!htb]
    \begin{tikzpicture}
        \draw[very thin,color=gray] (-4.1,-4.1) grid (4.1,4.1);
        \draw[->] (-4.2,0) -- (4.2,0) node[right] {$x$};
        \draw[->] (0,-4.2) -- (0,4.2) node[above] {$f(x)$};

        \draw[thick,purple,domain=-3:3,smooth] plot (\x,{0.5*\x^2});
        \draw[thick,brown,domain=-3:3,smooth] plot (\x,{0.2*\x^3});
    \end{tikzpicture}

\caption{$y=1/2 x^2, y=1/5 x^3$}
\end{figure}

\end{document}

Answer 1

The () are required so that exponents are properly applied to negative numbers. Otherwise we have -5^2=-25. Here is the output with and without the applied parenthesis:

Notes:

• Even though this will only be an issue with even exponents, I would recommend always using parenthesis (\x).

Code:

\documentclass[border=2pt]{standalone}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
        \draw[very thin,color=gray] (-4.1,-4.1) grid (4.1,4.1);
        \draw[->] (-4.2,0) -- (4.2,0) node[right] {$x$};
        \draw[->] (0,-4.2) -- (0,4.2) node[above] {$f(x)$};

        \draw[thick,purple,domain=-3:3,smooth] plot (\x,{0.5*\x^2}) node [right] {INCORRECT};
        \draw[thick,brown,domain=-3:3,smooth] plot (\x,{0.2*\x^3}) ;


    \begin{scope}[xshift=9cm]
        \draw[very thin,color=gray] (-4.1,-4.1) grid (4.1,4.1);
        \draw[->] (-4.2,0) -- (4.2,0) node[right] {$x$};
        \draw[->] (0,-4.2) -- (0,4.2) node[above] {$f(x)$};

        \draw[ultra thick,purple,domain=-3:3,smooth] plot (\x,{0.5*(\x)^2}) node [right] {CORRECT};
        \draw[ultra thick,brown,domain=-3:3,smooth] plot (\x,{0.2*(\x)^3});
    \end{scope}
\end{tikzpicture}
\end{document}