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\begin{document}
\section*{Homework 2 - Power Functions \\ 
 Due: February 21} 
\textbf{Instructions:} Please answer the following questions with well thought out answers. Write your answers on this paper. It is important to be able to explain ideas clearly. In this assignment pretend that you are trying to explain to a friend your answers. You should strive for your answers to be: precise, accurate, succinct, and understandable.
\vspace{.2 in} 

\begin{fdefn}
A \textbf{power function} has domain $\R$ and target $\R$ and is defined by $x \mapsto cx^r$, where $c$ and $r$ can be any fixed real numbers. 
\end{fdefn}

\vspace{.2 in}
\section*{Graphing by hand.} 
In the first half of the assignment we will focus on:  $f(x)=x$, $g(x)=x^2$, $h(x)=x^3$, $j(x)=x^4$, $k(x)=x^5$, and $l(x)=x^6$.  We will construct a table and use the table to sketch the graph of these functions. 
\begin{enumerate}
\item Consider the function $f: \R \to \R$ defined by $f(x)=x$.  
\begin{enumerate}
\item The function $f(x)=x$ is also known as the ``identity function."  Why do you think it is called that?\\
\vspace{.85 in}

\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
 \\
 \vspace{.9in}
\item Carefully sketch the graph of $f$, remembering that the domain is $\R$, and noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}


\end{multicols}

\newpage 

\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{.85in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $f$.  Is the inverse relation also a function?  
\vspace{.85in}

\item Write down one characteristic of the inverse that you notice.\\
\vspace{.85 in} 
\end{enumerate}

\item Consider the function $g: \R \to \R$ defined by $g(x)=x^2$.  
\begin{enumerate}

\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $g(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
 \\
 \vspace{.9in}
\item Carefully sketch the graph of $f$, remembering that the domain is $\R$, and noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}


\end{multicols}

\newpage 

\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{.85in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $f$.  Is the inverse relation also a function?  
\vspace{.85in}

\item Write down one characteristic of the inverse that you notice.\\
\vspace{.85 in} 


\item How is this graph similar and different from the previous function's graph?\\
\vspace{1in}
\end{enumerate}

\pagebreak

\item Consider the function $h: \R \to \R$ defined by $h(x)=x^3$.  
\begin{enumerate}


\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $h(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
  \\
 \vspace{.5in}
Then carefully sketch the graph of $h$, remembering that the domain is $\R$, noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}



\end{multicols}
\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{1in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $h$.  Is the inverse relation also a function?  Write down one characteristic of the inverse that you notice.\\
\vspace{1in}
\item How is this graph similar and different from the previous function's graph?\\
\vspace{1in}
\end{enumerate}

\pagebreak

\item Consider the function $j: \R \to \R$ defined by $j(x)=x^4$.  
\begin{enumerate}


\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $j(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
  \\
 \vspace{.5in}
Then carefully sketch the graph of $j$, remembering that the domain is $\R$, noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}



\end{multicols}
\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{1in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $j$.  Is the inverse relation also a function?  Write down one characteristic of the inverse that you notice.\\
\vspace{1in}
\item How is this graph similar and different from the previous function's graph?\\
\vspace{1in}
\end{enumerate}

\pagebreak

\item Consider the function $k: \R \to \R$ defined by $k(x)=x^5$.  
\begin{enumerate}


\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $k(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
  \\
 \vspace{.5in}
Then carefully sketch the graph of $k$, remembering that the domain is $\R$, noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}



\end{multicols}
\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{1in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $k$.  Is the inverse relation also a function?  Write down one characteristic of the inverse that you notice.\\
\vspace{1in}
\item How is this graph similar and different from the previous function's graph?\\
\vspace{1in}
\end{enumerate}

\pagebreak

\item Consider the function $l: \R \to \R$ defined by $l(x)=x^6$.  
\begin{enumerate}


\begin{multicols}{2}
\item Complete this table:\\
\begin{tabular}{|c|c|}
\hline
$x$ & $l(x)$\\ 
\hline 
&
\\ 
-3& \\ 
\hline&
\\ 
-2& \\ 
\hline 
&\\ 
-1& \\ 
\hline
&\\
$-\frac{1}{2}$ &\\
\hline
&\\
0&\\
\hline
&\\
$\frac{1}{2}$ &\\
\hline
&\\
1 & \\ 
\hline
&\\
2& \\
\hline
&\\
3 & \\
\hline

\end{tabular}
  \\
 \vspace{.5in}
Then carefully sketch the graph of $l$, remembering that the domain is $\R$, noting the scale, and including all points from your table.\\
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[help lines] (-9,-9) grid (9,9);
\draw[->] [ultra thick] (0,0) -- (0,9) ;
\draw[->] [ultra thick] (0,0) -- (9,0) ;

\draw[->] [ultra thick] (0,0) -- (-9,0) ;

\draw[->] [ultra thick] (0,0) -- (0,-9) ;
\end{tikzpicture}



\end{multicols}
\item Write down two characteristics of this function that you notice, and commit them to memory.\\
\vspace{1in}
\item On the same pair of axes above, sketch the graph of the {\bf{inverse relation}} of $l$.  Is its inverse also a function?  Write down one characteristic of the inverse that you notice.\\
\vspace{1in}
\item How is this graph similar and different from the previous function's graph?\\
\vspace{1in}
\end{enumerate}

\pagebreak

\item Describe what you think the graph of $m:\R \to \R$ defined by $m(x)=x^7$ looks like?  Why do you think that?  What about the power function $n(x)=x^8$?  Why?\\
\vspace{1in}

\item Label each of the power functions graphed below with one of the following: $f(x) =x, g(x) = x^2, h(x)=x^3,j(x)=x^4$.   \\
\begin{center}


\begin{tikzpicture}[xscale=1.5,yscale=1.5]
\draw[help lines] (-4,-4) grid (4,4);
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;

\draw[red, domain=-4:4] plot (\x, { \x});
\draw[green, domain=-2:2] plot (\x, { \x*\x});
\draw[blue, domain=-1.587:1.587] plot (\x, { \x*\x*\x});

\draw[orange, domain=-1.414:1.414] plot (\x, { \x*\x*\x*\x});

\end{tikzpicture}

\end{center}

\item What points do the functions have in common? \vspace{.8 in} 
\item For $0<x<1$, write $f(x),g(x),h(x),$ and $j(x)$ in ascending order.    \vspace{.8 in} 
\item For $x>1$,  write $f(x),g(x),h(x),$ and $j(x)$ in ascending order. \vspace{.8 in} 
\item When does a power function have an inverse relation that is also a function?  \vspace{.8 in} 
\item Describe any symmetry you see in the ``even" power functions.   \vspace{.8 in} 
\item Describe any symmetry you see in the ``odd" power functions. \vspace{.8 in} 


\end{enumerate}

\section*{Graphing using technology.}
In this section of the assignment you should make use of technology to help you graph the various functions. You can use a graphing calculator, or make use of one of the following the websites: 
\begin{center}  \url{http://www.wolframalpha.com}

\url{http://www.geogebra.org/cms/en/}
\end{center} 


\begin{enumerate} 

\item Use technology to sketch the graph of each of the following functions related to $f(x)=x^2$:
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^2$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = (x+2)^2$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = (x-2)^2$} ; 
\end{tikzpicture}
\end{center} 


\item Use technology to sketch the graph of each of the following functions related to $h(x)=x^3$:
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^3$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = (x+2)^3$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $h(x) = (x-2)^3$} ; 
\end{tikzpicture}
\end{center} 
\item Use Question 1 and Question 2 to make a conjecture about how the graphs of $f(x)=x^4, g(x) = (x+2)^4$ and $h(x) = (x-2)^4$ compare. \vspace{1 in} 
\item Decide if your conjecture is true by using technology to graph each of the following: 
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^4$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = (x+2)^4$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $h(x) = (x-2)^4$} ; 
\end{tikzpicture}
\end{center} 

\item Use technology to sketch the graph of each of the following functions related to $f(x)=x^2$:
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^2$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) =x^2+2$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = x^2-2$} ; 
\end{tikzpicture}
\end{center} 


\item Use technology to sketch the graph of each of the following functions related to $h(x)=x^3$:
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^3$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = x^3+2$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $h(x) = x^3-2$} ; 
\end{tikzpicture}
\end{center} 
\item Use Question 5 and Question 7 to make a conjecture about how the graphs of $f(x)=x^4, g(x) = x^4+2$ and $h(x) = x^4-2$ compare. \vspace{1 in} 
\item Decide if your conjecture is true by using technology to graph each of the following: 
\begin{center}
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^4$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $g(x) = x^4+2$} ; 
\end{tikzpicture}  \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $h(x) = x^4-2$} ; 
\end{tikzpicture}
\end{center} 
\item Given a function $f(x)$ make a conjecture about how the graphs for $f(x)$ and $-f(x)$ are related (hint: think about how the outputs are different). 
\item Test your conjecture Question 9 by using technology to graph the following graphs: 
\begin{center} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^4$} ; 
\end{tikzpicture} \hspace{.2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $-f(x) =-x^4$} ; 
\end{tikzpicture} 

\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $f(x) = x^3$} ; 
\end{tikzpicture} \hspace{ .2 in} 
\begin{tikzpicture}[xscale=.5,yscale=.5]
\draw[->] [ultra thick] (0,0) -- (0,4) ;
\draw[->] [ultra thick] (0,0) -- (4,0) ;
\draw[->] [ultra thick] (0,0) -- (-4,0) ;
\draw[->] [ultra thick] (0,0) -- (0,-4) ;
\draw (0, -5) node { $- f(x) =- x^3$} ; 
\end{tikzpicture}

\end{center} 
\end{enumerate} 

\end{document}


