Apply same options to multiple `addplot` in pgfplots

Just for future readers: at the time of writing this answer tikzstyle is deprecated. From the documentation

Starting with pgfplots 1.1, \tikzstyle should no longer be used to set pgfplots options. Although \tikzstyle is still supported for some older pgfplots options, you should replace any occurrence of \tikzstyle with \pgfplotsset{<style name>/.style={<key-value-list>}} or the associated /.append style variant

With this in mind

\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{tikz}

\pgfplotsset{style a/.style={color = red, mark = none, dashed}}

\begin{document}

\begin{tikzpicture}
   \begin{axis}[
        xmin=-10,
        xmax=10,
        xlabel={$x$},
        ylabel={$y$},
        ymin=-10,
        ymax=10]
     \addplot[style a] {x^2};
     \addplot[style a] {x^3 -4};
  \end{axis}
\end{tikzpicture}

\end{document}

You can define a style (or more):

\documentclass{standalone}
\usepackage{pgfplots}
\usepackage{tikz}

\tikzstyle{style_a}=[color = red, mark = none, dashed]

\begin{document}

\begin{tikzpicture}
   \begin{axis}[
        xmin=-10,
        xmax=10,
        xlabel={$x$},
        ylabel={$y$},
        ymin=-10,
        ymax=10]
     \addplot[style_a] {x^2};
     \addplot[style_a] {x^3 -4};
  \end{axis}
\end{tikzpicture}

\end{document}