#title: Grafico

#author: Irene and JJ

#let:

x[2] = {[-3..-1,1..3]};
y    = {[-2,-1,1,2]};

f  f0  = {{1/(x-#x[0]) + 1/(x-#x[1]) + #y}};
f  f2  = {{ratsimp(#f0)}};
s  f2s = #f2;

f  x_assin_hor = {{x,solve (1/(x-#x[0]) + 1/(x-#x[1]) + #y =#y)}};
f          y0  = {{ratsimp(1/(-#x[0]) + 1/(-#x[1]) + #y)}};

n xx_assin_hor = #x_assin_hor;
n yy0          = #y0;

#question:

\noindent Determine a função $f(x)=\frac{P(x)}{Q(x)},$ onde 
$P(x)$ e $Q(x)$ são polinómios de grau $\le 2$, corespondente ao
seguinte gráfico, sabendo que $f(0)= #y0$ e  $f(#x_assin_hor)=#y$;

\begin{tikzpicture}
     \begin{axis}[
       restrict y to domain=-7:7,
  %     grid,
       samples=1000,
       minor tick num=1,
       xlabel={$x$}, ylabel={$y$},
       xtickmax=6, ytickmax=5,
       xmin = -7, xmax = 7,
       ymin = -6, ymax = 6,
       unbounded coords=jump,
       axis x line=middle,
       axis y line=middle]
      \addplot [color=red,mark=none,domain=-7:7] {#f2s};
      \addplot [orange, no markers,dashed] coordinates {(#x[0],-6) (#x[0],6)};
      \addplot [orange, no markers,dashed] coordinates {(#x[1],-6) (#x[1],6)};
      \addplot [orange, no markers,dashed] coordinates {(-7,#y) (7,#y)};
      \addplot [black, mark=*, only marks] coordinates {(0,#yy0) (#xx_assin_hor,#y)};
       \node at (axis cs:0,#yy0          ) [anchor=north east] {$A$};
       \node at (axis cs:#xx_assin_hor,#y) [anchor=north east] {$B$};
      \end{axis}
\end{tikzpicture}

#sugestion:
#resolution: 
 
$$
 f(x) = #f0 = #f2
$$

#result:

#usepackage:
 
\usepackage{pgfplots}
\usepackage{tikz}