Resources for Curve Sketching
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Questions
7
With Worked SolutionClick Here -
Video Tutorials
1
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HSC Questions
2
With Worked SolutionClick Here
Curve Sketching Theory
![Curve sketching requires all the information that has been stated in the question to appear on the sketch.\\ \begin{multicols}{2} \textbf{Example 1}\\ Sketch the curve \(y=x^3-6 x^2+9 x+5\) showing the coordinates of maximum and minimum turning points, coordinates of points of inflexion and the intercept on the \(y\)-axis.\\ \textbf{Example 1 solution}\\ $\begin{aligned} & y=x^3-6 x^2+9 x+5 \\ & y^{\prime}=3 x^2-12 x+9 \\ & y^{\prime \prime}=6 x-12 \end{aligned}$\\ $\begin{aligned} \text { Let } y^{\prime}=0 \quad &3 x^2-12 x+9=0 \\ &x^2-4 x+3=0 \\ &(x-3)(x-1)=0 \\ &\therefore x=3 \text { or } x=1 \end{aligned}$\\ $\begin{aligned} \text { At } x=3 \quad &y =3^3-6 \times 3^2+9 \times 3+5 \\ &y =5 \\ \text { At } x=1 \quad &y=1^3-6 \times 1^2+9 \times 1+5 \\ &y =9 \end{aligned}$\\ At \(x=3 \quad y^{\prime \prime}>0 \quad \therefore(3,5)\) is a minimum stationary point.\\ At \(x=1 \quad y^{\prime \prime}<0 \quad\therefore(1,9)\) is a maximum stationary point.\\ Let \(y^{\prime \prime}=0\)\\ $\begin{aligned} 6 x-12 =0& \\ x =2& \quad y=2^3-6 \times 2^2+9 \times 2+5 \\ &\quad y =7 \end{aligned}$\\ \columnbreak \textbf{Solution continued}\\ \(\therefore(2,7)\) is a point of inflexion as it lies between a maximum and a minimum turning point.\\ At \(x=0 \quad y=5\)\\ \(\therefore(0,5) \text { is the y-intercept. }\)\\ \begin{center} \begin{tikzpicture}[ declare function={a(\x)=(\x)^3-6*\x^2+9*\x+5;}, ] \def \domain{-0.47:4.43} \def \xmax{6} \def \xmin{-3} \def \ymax{14} \def \ymin{-1} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={Stealth-Stealth,very thick}, grid=both, %major, ylabel = $\ylabel$, xlabel = $\xlabel$, width=3.8in, height=3.5in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, minor x tick num=1, minor y tick num=1, axis line style = thick, major tick style = thick, minor tick style = thick, xtick distance = 1, xlabel style={right}, ytick distance = 2, ylabel style={above}, x grid style={thin, opacity=0.8}, y grid style={thin, opacity=0.8}, axis on top=false, xtick={-3,...,6}, ytick={-2,0,2,4,...,14}, % extra x ticks={0}, extra x tick style={xticklabel style={anchor=north east}} ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax); \addplot[name path=a, ultra thick, latex-latex, samples=300, smooth, domain=\domain, red] {a(x)} node [pos=0.9, left, red, font=\small] {}; \node (11) at (1,9) {}; \node (33) at (0,5) {}; \node (22) at (3,5) {}; \node (44) at (2,7) {}; \node (1) [draw ,rectangle, align=center,fill=gray!20] at (-0.6,2) { Local Maximum\\ \((1,9)\)}; \node (2) [draw ,rectangle, align=center,fill=gray!20] at (4,2) { Local Minimum\\ \((3,5)\)}; \node (3) [draw ,rectangle, align=center,fill=gray!20] at (-1.5,10) {\(y\) Intercept\\\((0,5)\)}; \node (4) [draw ,rectangle, align=center,fill=gray!20] at (3,11.5) {Point of Inflection\\ \((2,7)\)}; \end{axis} \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (1)--(11); \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (2)--(22); \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (3)--(33); \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (4)--(44); \end{tikzpicture} %\includegraphics[width=0.45\textwidth]{Picture100} \end{center} \end{multicols}](/media/jatjba4o/4772.png)