Problem source

Let
$$y^2=x^2(a^2-x^2),$$
where $a$ is a real constant.
Find, in terms of $a$, the maximum and minimum values of $y$.

Sketch carefully on the same axes the graphs of $y$
in the cases $a=1$ and $a=2$.

Solution source

\begin{align*}
&& y^2 &= x^2a^2-x^2 \\
&&&= \frac{a^4}{4} -\left ( x^2 -\frac{a^2}{2} \right)^2
\end{align*}

Therefore the maximum and minimum values of $y$ are $\pm \frac{a^2}2$



\begin{center}
    \begin{tikzpicture}
    \def\a{-0.8};
    \def\functionf(#1){(#1)^2*(4-(#1)^2)};
    \def\functiong(#1){(#1)^2*(1-(#1)^2)};
    \def\xl{-3};
    \def\xu{3};
    \def\yl{-3};
    \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[blue, smooth, thick, domain=-2:2, samples=101]
            plot({\x}, {sqrt(\functionf(\x))}) -- (2,0); 
        \draw[blue, smooth, thick, domain=-2:2, samples=101]
            plot({\x}, {-sqrt(\functionf(\x))}) -- (2,0); 
        \draw[red, smooth, thick, domain=-1:1, samples=101]
            plot({\x}, {sqrt(\functiong(\x))}) -- (1,0); 
        \draw[red, smooth, thick, domain=-1:1, samples=101]
            plot({\x}, {-sqrt(\functiong(\x))}) -- (1,0); 
        % \draw[red, smooth, thick, domain=1:2, samples=101]
            % plot({\x}, {1+3-1}); 
        % \draw[red, smooth, thick, domain=2:3, samples=101]
            % plot({\x}, {4+6-1}); 
        
        \node[blue, above, rotate=60] at (0.75, {1.3}) {\tiny $y^2=x^2(4-x^2)$};

        \filldraw (1, 0) circle (1.5pt) node[below] {\tiny $1$};
        \filldraw (-1, 0) circle (1.5pt) node[below] {\tiny $-1$};
        \filldraw (2, 0) circle (1.5pt) node[below] {\tiny $2$};
        \filldraw (-2, 0) circle (1.5pt) node[below] {\tiny $-2$};
        
        \filldraw ({sqrt(2)}, 2) circle (1.5pt) node[above] {\tiny $(\sqrt{2},2)$};
        \filldraw ({-sqrt(2)}, 2) circle (1.5pt) node[above] {\tiny $(-\sqrt{2},2)$};
        \filldraw ({sqrt(2)}, -2) circle (1.5pt) node[below] {\tiny $(\sqrt{2},-2)$};
        \filldraw ({-sqrt(2)}, -2) circle (1.5pt) node[below] {\tiny $(-\sqrt{2},-2)$};

        \filldraw ({1/sqrt(2)}, 0.5) circle (1.5pt) node[above] {\tiny $(\frac{1}{\sqrt{2}},\frac12)$};
        \filldraw ({-1/sqrt(2)}, 0.5) circle (1.5pt) node[above] {\tiny $(-\frac{1}{\sqrt{2}},\frac12)$};
        \filldraw ({1/sqrt(2)}, -0.5) circle (1.5pt) node[below] {\tiny $(\frac{1}{\sqrt{2}},-\frac12)$};
        \filldraw ({-1/sqrt(2)}, -0.5) circle (1.5pt) node[below] {\tiny $(-\frac{1}{\sqrt{2}},-\frac12)$};
        % \filldraw (-1, 0) circle (1.5pt) node[below] {$-1$};
        % \filldraw ({-2/3}, {\functionf(-2/3)}) circle (1.5pt) node[below] {$(-\frac23, -\frac{4078}{81})$};
        \filldraw ({1/sqrt(2)}, {243/64}) circle (1.5pt) node[above] {\tiny $(\frac1{\sqrt{2}},\frac{243}{64})$};
        \filldraw ({-1/sqrt(2)}, {243/64}) circle (1.5pt) node[above] {\tiny $(-\frac1{\sqrt{2}},\frac{243}{64})$};
    \end{scope}

    
    % 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}