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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
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\lhead{\color{blue}\large  Lecture Notes}
\chead{{\color{black}}{\Large{Graphs of Equations}}}
\rhead{\small page   \ \thepage}
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\lfoot{\footnotesize   \copyright $\;$   Hidegkuti,  Powell,  2007}
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\begin{document}


Definition: \ The graph of an equation in $x$ and $y$ is the set of all
points $P\left( x,y\right) $ \ whose coordinates are solution of the
equation.

\begin{enumerate}
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Among the many points on the graph, \ $A\left( -2,0\right) $, \ $B\left(
-1,5\right) $, \ and $C\left( 1,9\right) $ \ are marked \ \ We will use
these points to find the equation that belongs with the graph. \ The
equation for this graph is one of the equations listed below.

\begin{enumerate}
\item Consider the equation $y=3x+6$.

\begin{enumerate}
\item Is the point $A\left( -2,0\right) $ \ on the graph of $y=3x+6$?

\item Is the point $B\left( -1,5\right) $ \ \ on the graph of $y=3x+6$?

\item Is the point $C\left( 1,9\right) $ \ on the graph of $y=3x+6$?

\item Is it possible that the graph shown on the picture above is the graph
of $y=3x+6$?
\end{enumerate}

\item Consider the equation \ $\left( x-4\right) ^{2}+\left( y-5\right)
^{2}=25$

\begin{enumerate}
\item Is the point $A\left( -2,0\right) $ \ on the graph of $\left(
x-4\right) ^{2}+\left( y-5\right) ^{2}=25$?

\item Is the point $B\left( -1,5\right) $ \ \ on the graph of $\left(
x-4\right) ^{2}+\left( y-5\right) ^{2}=25$?

\item Is the point $C\left( 1,9\right) $ \ on the graph of $\left(
x-4\right) ^{2}+\left( y-5\right) ^{2}=25$?

\item Is it possible that the graph shown on the picture above is the graph
of

$\left( x-4\right) ^{2}+\left( y-5\right) ^{2}=25$?
\end{enumerate}

\item Consider the equation \ $y=-x^{2}+2x+8$.

\begin{enumerate}
\item Is the point $A\left( -2,0\right) $ \ on the graph of $y=-x^{2}+2x+8$?

\item Is the point $B\left( -1,5\right) $ \ \ on the graph of $y=-x^{2}+2x+8$%
?

\item Is the point $C\left( 1,9\right) $ \ on the graph of $y=-x^{2}+2x+8$?

\item Is it possible that the graph shown on the picture above is the graph
of $y=-x^{2}+2x+8$?
\end{enumerate}
\end{enumerate}

\pagebreak 

\item Consider the graph shown on the picture below.

Note that the points $A\left( -8,1\right) $, \ $B\left( -6,5\right) $, \ \
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\begin{enumerate}
\item Consider the equation \ $3y=x+11$.

\begin{enumerate}
\item Is the point $A\left( -8,1\right) $ \ on the graph of $3y=x+11$?~

\item Is the point $B\left( -6,5\right) $ \ \ on the graph of $3y=x+11$?

\item Is the point $C\left( 1,4\right) $ \ on the graph of $3y=x+11$?

\item Is it possible that the graph shown on the picture above is the graph
of $3y=x+11$?\bigskip 
\end{enumerate}

\item Consider the equation \ $3y+x^{2}=-8x+3$.

\begin{enumerate}
\item Is the point $A\left( -8,1\right) $ \ on the graph of $3y+x^{2}=-8x+3$?

\item Is the point $B\left( -6,5\right) $ \ \ on the graph of $3y+x^{2}=-8x+3
$?

\item Is the point $C\left( 1,4\right) $ \ on the graph of $3y+x^{2}=-8x+3$?

\item Is it possible that the graph shown on the picture above is the graph
of $3y+x^{2}=-8x+3$?\bigskip 
\end{enumerate}

\item Consider the equation \ $\left( x+3\right) ^{2}+\left( y-1\right)
^{2}=25$.

\begin{enumerate}
\item Is the point $A\left( -8,1\right) $ \ on the graph of $\left(
x+3\right) ^{2}+\left( y-1\right) ^{2}=25$?

\item Is the point $B\left( -6,5\right) $ \ \ on the graph of $\left(
x+3\right) ^{2}+\left( y-1\right) ^{2}=25$?

\item Is the point $C\left( 1,4\right) $ \ on the graph of $\left(
x+3\right) ^{2}+\left( y-1\right) ^{2}=25$?

\item Is it possible that the graph shown on the picture above is the graph
of

$\left( x+3\right) ^{2}+\left( y-1\right) ^{2}=25$?\pagebreak 
\end{enumerate}
\end{enumerate}

\item Consider the graph shown below.

Note that the points \ $A\left( -3,4\right) $, \ $B\left( 0,-2\right) $, \
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\begin{enumerate}
\item Consider the equation \ $y+2=\left\vert 2x\right\vert $.

\begin{enumerate}
\item Is the point $A\left( -3,4\right) $ \ on the graph of $y+2=\left\vert
2x\right\vert $?

\item Is the point $B\left( 0,-2\right) $ \ \ on the graph of $%
y+2=\left\vert 2x\right\vert $?

\item Is the point $C\left( 7,0\right) $ \ on the graph of $y+2=\left\vert
2x\right\vert $?

\item Is it possible that the graph shown on the picture above is the graph
of $y+2=\left\vert 2x\right\vert $?\bigskip 
\end{enumerate}

\item Consider the equation $\ 6-y=\left\vert 8-\left\vert 2x\right\vert
\right\vert $.

\begin{enumerate}
\item Is the point $A\left( -3,4\right) $ \ on the graph of $6-y=\left\vert
8-\left\vert 2x\right\vert \right\vert $?

\item Is the point $B\left( 0,-2\right) $ \ \ on the graph of $%
6-y=\left\vert 8-\left\vert 2x\right\vert \right\vert $?

\item Is the point $C\left( 7,0\right) $ \ on the graph of $6-y=\left\vert
8-\left\vert 2x\right\vert \right\vert $?

\item Is it possible that the graph shown on the picture above is the graph
of $6-y=\left\vert 8-\left\vert 2x\right\vert \right\vert $?\bigskip 
\end{enumerate}

\item Consider the equation \ $x^{2}+y^{2}=1+4\left( x+y+5\right) $.

\begin{enumerate}
\item Is the point $A\left( -3,4\right) $ \ on the graph of $%
x^{2}+y^{2}=1+4\left( x+y+5\right) $?

\item Is the point $B\left( 0,-2\right) $ \ \ on the graph of $%
x^{2}+y^{2}=1+4\left( x+y+5\right) $?

\item Is the point $C\left( 7,0\right) $ \ on the graph of $%
x^{2}+y^{2}=1+4\left( x+y+5\right) $?

\item Is it possible that the graph shown on the picture above is the graph
of

$x^{2}+y^{2}=1+4\left( x+y+5\right) $?\pagebreak 
\end{enumerate}
\end{enumerate}

\item Consider the graph shown on the picture below.

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\begin{enumerate}
\item Consider the equation \ $2y=x+9$.

\begin{enumerate}
\item Is the point $A\left( -3,3\right) $ \ on the graph of $2y=x+9$?

\item Is the point $B\left( 1,5\right) $ \ \ on the graph of $2y=x+9$?

\item Is the point $C\left( 4,-2\right) $ \ on the graph of $2y=x+9$?~

\item Is it possible that the graph shown on the picture above is the graph
of $2y=x+9$?\bigskip 
\end{enumerate}

\item Consider the equation \ $\left\vert x\right\vert +\left\vert
y\right\vert =6$.

\begin{enumerate}
\item Is the point $A\left( -3,3\right) $ \ on the graph of $\left\vert
x\right\vert +\left\vert y\right\vert =6$?

\item Is the point $B\left( 1,5\right) $ \ \ on the graph of $\left\vert
x\right\vert +\left\vert y\right\vert =6$?

\item Is the point $C\left( 4,-2\right) $ \ on the graph of $\left\vert
x\right\vert +\left\vert y\right\vert =6$?

\item Is it possible that the graph shown on the picture above is the graph
of $\left\vert x\right\vert +\left\vert y\right\vert =6$?\bigskip 
\end{enumerate}

\item Consider the equation $\ y+3=9-\left\vert x\right\vert $.

\begin{enumerate}
\item Is the point $A\left( -3,3\right) $ \ on the graph of $%
y+3=9-\left\vert x\right\vert $?

\item Is the point $B\left( 1,5\right) $ \ \ on the graph of $%
y+3=9-\left\vert x\right\vert $?

\item Is the point $C\left( 4,-2\right) $ \ on the graph of $%
y+3=9-\left\vert x\right\vert $?~

\item Is it possible that the graph shown on the picture above is the graph
of $y+3=9-\left\vert x\right\vert $?
\end{enumerate}
\end{enumerate}
\end{enumerate}

\pagebreak 

\begin{center}
{\Large Answers}\bigskip 
\end{center}

\begin{enumerate}
\item a)  i) \ $0=0~~~$yes \ \ \ \ \ \ \ ii) \ $5\not=3~~~$no \ \ \ \ \ iii)
\ $9=9~~~$yes \ \ \ \ \ \ iv) \ no

b) \ i) \ $36\not=25~~\ $no \ \ \ \ \ ii) \ $25=25~~~$yes \ \ \ \ \ \ iii) \ 
$25=25~~~$yes \ \ \ \ \ \ \ iv) \ no

c) \ i) \ $0=0$ \ \ yes \ \ \ \ ii) \ $5=5$ \ yes \ \ \ \ \ iii) \ $9=9$ \ \
yes \ \ \ \ iv) \ yes

\item a) \ i) $\ 3=3$ yes \ \ \ ii) \ $15\not=5$ \ \ no \ \ \ \ \ iii) \ $%
12=12$ \ yes \ \ \ \ iv) \ no

b) \ i) \ \ $67=67$ \ yes \ \ \ \ ii) \ $51=51$ \ yes \ \ \ \ \ iii) \ $%
13\not=-5$ \ \ \ \ \ iv) \ no

c) \ i) \ $25=25$ yes \ \ \ \ ii) \ $25=25$ \ yes \ \ \ \ iii) \ $25=25$ \
yes \ \ \ \ iv) \ yes

\item a) \ i) \ $6=6$ \ \ yes \ \ \ \ ii) \ $2=2$ \ yes \ \ \ \ \ iii) \ \ $%
2\not=14$ \ \ no \ \ \ \ \ \ iv) \ no

b) \ i) $\ 2=2$ \ yes \ \ \ \ ii) \ $8=8$ \ yes \ \ \ \ iii) \ \ $6=6$ \ yes
\ \ \ \ iv) \ yes

c) \ i) \ $25=25$ \ yes \ \ \ ii) \ $4\not=13$ \ no \ \ \ \ iii) \ $49=49$ \
yes \ \ iv) \ no

\item a) \ i) \ $6=6$ \ yes \ \ \ \ ii) \ $10=10$ \ yes \ \  \ \ \ \ iii) \ $%
-4\not=13$ \ no \ \ \ \ \ iv) \ no

b) \ i) \ $6=6$ \ yes \ \ \ ii) \ $6=6$ \ yes \ \ \ \ \ iii) \ $6=6$ \ yes \
\ \ \ iv) \ yes

c) \ i) \ $6=6$ \ yes \ \ \ ii) \ $8=8$ \ yes \ \ \ \ iii) \ $1\not=5$ \ no
\ \ \ \ \ iv) \ no
\end{enumerate}

\end{document}