Exercise 1.2.10

Question

Sketch several members of the family of polynomials $P(x) =x^3 - cx^2$. How does the graph change when $c$ changes?

Khagan Gulusoy · Accepted Answer

Let's take $c=2,$ $c=1,$ $c=-2.$ The blue plot illustrates the case when $y$-intercept is equal to $2;$ the red plot - when $y$-intercept is equal to $1$; the green plot - when $y$-intercept equal to $-2.$

\begin{figure}[H] \centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, xmin=-3, xmax=3, ymax=4, ymin=-4, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-4:4, samples=50, mark=none, color=blue, ultra thick] {x^3-2*x^2};
         \addplot[domain=-4:4, samples=50, mark=none, color=red, ultra thick] {x^3-x^2};
         \addplot[domain=-4:4, samples=50, mark=none, color=green!60!black, ultra thick] {x^3+2*x^2};
    \end{axis}    
    \end{tikzpicture}
\end{figure}
%The red plot illustrates the function of the given family with $y$-intercept equal to $1$;

We can use graphing software to plot these functions:
\begin{itemize}
\item Using Desmos: click on \href{https://www.desmos.com/calculator/qlm7f2y27z?embed}{this link}.
\item Using WolframAlpha: click on \href{https://www.wolframalpha.com/input?i=plot+x%5E3-2*x%5E2%2C+x%5E3-x%5E2%2C+x%5E3%2C+x%5E3%2Bx%5E2%2C+x%5E3%2B2*x%5E2%2C+x%5E3%2B3*x%5E2%2Cx%3D-4+to+4%2C+y%3D-4+to+4}{this link}.
\end{itemize}

Exercise 1.2.10

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Exercise 1.2.10

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