如何绘制具有垂直和斜渐近线的曲线？

Question

我怎样才能绘制一条具有垂直和斜渐近线的曲线，如下图所示？

曲线方程： y=\sqrt[3]{\frac{x^7+3x^6-2x^5-x^4-3x^2+1}{x^4-2x^2}}

Answer 1

当然，你可以用其他软件来画图，这确实有一些优势。但你也可以用 LaTeX 来画图，这样你就不用担心添加方程式了。这里是对该主题的延伸讨论。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}[declare function={%
    f(\x)=pow((x^7+3*x^6-2*x^5-x^4-3*x^2+1)/(x^4-2*x^2),1/3);}]
  \begin{axis}[xmin=0,xmax=5,ymin=-0.5,ymax=6,xlabel={$x$},ylabel={$y$},grid=major,
   title={$\displaystyle f(x)=\sqrt[3]{\frac{x^7+3x^6-2x^5-x^4-3x^2+1}{x^4-2x^2}}$}]
    \addplot[color=blue,semithick,domain={sqrt(2.035)}:5,samples=71] {f(x)}
    node[pos=0.75,above,sloped]{$y=f(x)$};
    \addplot[color=blue,semithick,domain=0.55:1.11,samples=71] {f(x)};
    \addplot[color=green!70!black,domain=0:5,samples=2] {x}
    node[midway,below,sloped]{$y=x$};
    \addplot[color=green!70!black,domain=-0.5:6,samples=2] ({sqrt(2)},{x})
    node[midway,above,sloped]{$x=\sqrt{2}$};
    \addplot[color=red!70!black,domain=0:5,samples=2] {1.075};
 \end{axis}
\end{tikzpicture}
\end{document}

Answer 2

解决方案如下pstricks：

\documentclass[11pt,svgnames, border=5pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{fourier}

\usepackage{pst-plot, multido}
\usepackage{auto-pst-pdf}
\def\f{((x^7 + 3*x^6-2*x^5-x^4-3*x^2 + 1)/(x^4-2*x^2))^(1/3)}
\def\g{-((x^7 + 3*x^6-2*x^5-x^4-3*x^2 + 1)/(2*x^2- x^4))^(1/3)}

\begin{document}

\psset{plotstyle=curve, algebraic, arrowinset=0.15, dx=2,Dx = 2}%
\psset{subgriddiv=2,gridcolor=LightSteelBlue, subgridcolor=LightSteelBlue!40, yunit=1.25}
\sffamily
\begin{pspicture}(-0.5,-0.8)(8.5, 6.5)
\psclip{\psframe(-0.5,-0.8)(8.5,6.5)}
\psgrid[xunit=2, yunit=2, gridlabels=0pt, subgriddiv = 4]
\psaxes[labels=none, arrows=->, arrowscale=1.3, xsubticks=4, ysubticks=2, subtickcolor=black](0,0)(7.5,5.8)[$x$,-130] [$y$,-40]%
\uput[r](0,2){2.688}\uput[r](0,4){5.375}\uput{12pt}[l](8.5,2){2.688}\uput{12pt}[l](8.5,4){5.375}
 \psplot[plotpoints=10, linewidth=1.2pt, linecolor=DodgerBlue]{0.549631}{1.103438}{\f}
 \psplot[plotpoints=10, linewidth=1.2pt, linecolor=DodgerBlue]{0.41}{0.54963}{\g}
 \psplot[plotpoints=10, linewidth=1.2pt, linecolor=DodgerBlue]{1.103439}{1.414}{\g}
\psplot[plotpoints=200, linewidth=1.2pt, linecolor=DodgerBlue]{1.418}{8}{\f}
\psline[linecolor=LightSeaGreen](1.4,-1)(1.4,8)
\psline[linecolor=LightSeaGreen](0,0)(8,8)
\multido{\n = -0.5 + 0.5}{16}{\psline[linewidth=0.3pt](\n, -0.8)(\n, -0.65)\psline[linewidth=0.3pt](\n, 6.5)(\n, 6.35)%
 \psline[linewidth=0.3pt](-0.5, \n)(-0.35,\n) \psline[linewidth=0.3pt](8.5, \n)(8.35,\n)}
\multido{\i = 0 + 2}{5}{\uput{10pt}[d](\i, 0){\i}\psline(\i, -0.8)(\i, -0.6)%
\psline(-0.5,\i)(-0.25,\i)\psline(8.5,\i)(8.25,\i)
\uput{11pt}[d](\i, 6.5){\i}\psline(\i, 6.5)(\i,6.3)}
\endpsclip
\end{pspicture}

\end{document}