Problem source

Sketch the following subsets of the $x$-$y$ plane:
\begin{questionparts}
\item $|x|+|y|\le 1$ ;
\item $|x-1|+|y-1|\le 1 $ ;
\item $|x-1|-|y+1|\le 1 $ ;
\item $|x|\, |y-2|\le 1$ .
\end{questionparts}

Solution source

\begin{questionparts}

\item 

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){sin(deg(#1))};
    \def\xl{-1.5};
    \def\xu{1.5};
    \def\yl{-1.5};
    \def\yu{1.5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \filldraw[red, opacity=0.5] (1,0) -- (0,1) -- (-1, 0) -- (0, -1) -- cycle;
        \draw[thick] (1,0) -- (0,1) -- (-1, 0) -- (0, -1) -- cycle;

    \end{scope}

    \node[below] at (1,0) {$(1,0)$};
    \node[left] at (0,1) {$(0,1)$};
    \node[left] at (0,-1) {$(0,-1)$};
    \node[below] at (-1,0) {$(-1,0)$};
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}


\item 
\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){sin(deg(#1))};
    \def\xl{-1};
    \def\xu{3};
    \def\yl{-1};
    \def\yu{3};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \filldraw[red, opacity=0.5] (1,0) -- (0,1) -- (1, 2) -- (2, 1) -- cycle;
        \draw[thick] (1,0) -- (0,1) -- (1, 2) -- (2, 1) -- cycle;
    \end{scope}

    \node[below] at (1,0) {$(1,0)$};
    \node[left] at (0,1) {$(0,1)$};
    \node[above] at (1,2) {$(1,2)$};
    \node[right] at (2,1) {$(2,1)$};
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}


\item 
\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){sin(deg(#1))};
    \def\xl{-5};
    \def\xu{5};
    \def\yl{-5};
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \filldraw[red, opacity=0.5] (20,19) -- (2,-1) -- (20,-21)  -- cycle;
        \filldraw[red, opacity=0.5] (-20,19) -- (0,-1) -- (-20,-21)  -- cycle;
        % \draw[thick] (1,0) -- (0,1) -- (1, 2) -- (2, 1) -- cycle;
    \end{scope}

    \node[above] at (2,-1) {$(2,-1)$};
    \node[below] at (0,-1) {$(0,-1)$};
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}


\item \begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){sin(deg(#1))};
    \def\xl{-8};
    \def\xu{8};
    \def\yl{-8};
    \def\yu{8};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \filldraw[red, opacity=0.5, domain=0.05:5, samples=1000, variable=\x]
          plot ({1/\x}, {\x+2}) 
          -- plot({-\x}, {1/\x+2})
          -- plot({-1/\x}, {-(\x)+2})
          -- plot({\x}, {-(1/\x)+2})
          -- cycle;
        \draw[red, thick, domain=0.05:5, samples=1000, variable=\x]
          plot ({1/\x}, {\x+2}) 
          -- plot({-\x}, {1/\x+2})
          -- plot({-1/\x}, {-(\x)+2})
          -- plot({\x}, {-(1/\x)+2})
          -- cycle;
        \draw[red,dashed] (-20,2) -- (20, 2);
        % \filldraw[red, opacity=0.5] (-20,19) -- (0,-1) -- (-20,-21)  -- cycle;
        % \draw[thick] (1,0) -- (0,1) -- (1, 2) -- (2, 1) -- cycle;
    \end{scope}

    \node at (0,2) {$(0,2)$};
    % \node[below] at (0,-1) {$(0,-1)$};
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}



\end{questionparts}