1990 Paper 2 Q3
Year: 1990
Paper: 2
Question Number: 3
Course: LFM Stats And Pure
Section: Curve Sketching
Difficulty: 1600.0
Banger: 1500.0
curve sketching
cubic polynomial
stationary points
discriminant
parametric family
implicit curve
factorisation
Problem
Sketch the curves given by
\[
y=x^{3}-2bx^{2}+c^{2}x,
\]
where \(b\) and \(c\) are non-negative, in the cases:
\begin{questionparts}
\item \(2b < c\sqrt{3}\)
\item \(2b=c\sqrt{3}\neq0\)
\item \(c\sqrt{3} < 2b < 2c\),
\item \(b=c\neq0\)
\item \(b > c > 0\),
\item \(c=0,b\neq0\)
\item \(c=b=0\).
\end{questionpart}
Sketch also the curves given by \(y^{2}=x^{3}-2bx^{2}+c^{2}x\) in the
cases \((i), (v)\) and \((vii)\).
Solution
Rating Information
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Show LaTeX source
Problem source
Sketch the curves given by
\[
y=x^{3}-2bx^{2}+c^{2}x,
\]
where $b$ and $c$ are non-negative, in the cases:
\begin{questionparts}
\item $2b < c\sqrt{3}$
\item $2b=c\sqrt{3}\neq0$
\item $c\sqrt{3} < 2b < 2c$,
\item $b=c\neq0$
\item $b > c > 0$,
\item $c=0,b\neq0$
\item $c=b=0$.
\end{questionpart}
Sketch also the curves given by $y^{2}=x^{3}-2bx^{2}+c^{2}x$ in the
cases $\textbf{(i)}, \textbf{(v)}$ and $\textbf{(vii)}$.
Solution source
\begin{questionparts}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*1*(#1)^2+4/2*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*1*(#1)^2+4/3*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*(#1)^2+7/6*4*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*(#1)^2+4*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*(#1)^2+3*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*(#1)^2+(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*(#1)^2+0*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\item \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*0*(#1)^2+0*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[thick, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\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}
\end{questionparts}
$\textbf{(i)}$ \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*1*(#1)^2+4/2*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[dashed, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, blue, smooth, domain=0:5, samples=100]
plot (\x, {sqrt(\functionf(\x))});
\draw[thick, blue, smooth, domain=0:5, samples=100]
plot (\x, {-sqrt(\functionf(\x))});
\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}
$\textbf{(v)}$ \begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*1.05*(#1)^2+4*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[dashed, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, blue, smooth, domain=2.744:5, samples=100]
plot (\x, {sqrt(\functionf(\x))});
\draw[thick, blue, smooth, domain=2.744:5, samples=100]
plot (\x, {-sqrt(\functionf(\x))});
\draw[thick, blue, smooth, domain=0:1.458, samples=100]
plot (\x, {sqrt(\functionf(\x))});
\draw[thick, blue, smooth, domain=0:1.458, samples=100]
plot (\x, {-sqrt(\functionf(\x))});
% \draw[thick, blue, smooth, domain=-5:0, samples=100]
% plot (\x, {sqrt(\functionf(\x))});
% \draw[thick, blue, smooth, domain=-5:0, samples=100]
% plot (\x, {-sqrt(\functionf(\x))});
\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}
$\textbf{(vii)}$
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){(#1)^3-2*2*0*(#1)^2+0*(#1)};
% \def\functiong(#1){((#1)+5)*((#1)-4)*((#1)+1)/((#1)-5)/((#1)+4)};
\def\xl{-5};
\def\xu{5};
\def\yl{-10};
\def\yu{10};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{1}
\pgfmathsetmacro{\yscale}{\xrange/\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);
\draw[dashed, blue, smooth, domain=-5:5, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, blue, smooth, domain=0:5, samples=100]
plot (\x, {sqrt(\functionf(\x))});
\draw[thick, blue, smooth, domain=0:5, samples=100]
plot (\x, {-sqrt(\functionf(\x))});
% \draw[thick, blue, smooth, domain=-5:0, samples=100]
% plot (\x, {sqrt(\functionf(\x))});
% \draw[thick, blue, smooth, domain=-5:0, samples=100]
% plot (\x, {-sqrt(\functionf(\x))});
\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}