\hfill \hfill } %} \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} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.7in} \pagestyle{fancy} \lhead{\color{blue}\LARGE } \chead{{\color{black}}{\LARGE{Graphs of Equations}}} \rhead{\large page \thepage} \cfoot{} \lfoot{\small \copyright $\;$ copyright Hidegkuti, Powell, 2007} \textwidth 7.0in \textheight 9.0in \setlength{\headheight}{38pt} \setlength{\parindent}{0in} \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.\bigskip \begin{enumerate} \item Consider the graph shown below. \FRAME{dtbpF}{3.5129in}{3.5129in}{0pt}{% }{}{Plot}{\special{language "Scientific Word";type "MAPLEPLOT";width 3.5129in;height 3.5129in;depth 0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin "-10.01";xmax "10.04";xviewmin "-10.01";xviewmax "10.04";yviewmin "-10.4";yviewmax "10.4";viewset"XY";rangeset"X";plottype 4;plotticks 1;num-x-ticks 11;num-y-ticks 11;constrained 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"[flat::RGB:0x000000ff]";curveStyle "Line";rangeset"X";valid_file "T";tempfilename 'JOUCXY03.wmf';tempfile-properties "XPR";}}Note that the points \ $A\left( -4,1\right) $,\ \ $B\left( 2,4\right) $, \ $C\left( 4,5\right) $, \ and \ $D\left( -1,-5\right) $ \ are on the graph.\bigskip Determine which of the following equations belongs to this graph.\bigskip a) \ $8y=5x^{2}+14x-16$ b) \ $2y-3=x+3$ c) \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $% \pagebreak \begin{enumerate} \item Consider the equation \ $8y=5x^{2}+14x-16$ \ and points $A\left( -4,1\right) $,\ \ $B\left( 2,4\right) $, \ and \ $C\left( 4,5\right) .$% \bigskip \begin{enumerate} \item Is the point $A\left( -4,1\right) $ \ on the graph of \ $% 8y=5x^{2}+14x-16$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=-4$ \ and $y=1$. \ Is this pair a solution of $\ 8y=5x^{2}+14x-16$?% \begin{eqnarray*} \text{LHS} &=&8\left( 1\right) =8 \\ \text{RHS} &=&5\left( -4\right) ^{2}+14\left( -4\right) -16=5\cdot 16+14\left( -4\right) -16=80-56-16 \\ &=&24-16=8 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of $8y=5x^{2}+14x-16$. \item Is the point $A\left( 2,4\right) $ \ on the graph of \ $% 8y=5x^{2}+14x-16$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=2$ \ and $y=4$. \ Is this pair a solution of $8y=5x^{2}+14x-16$?% \begin{eqnarray*} \text{LHS} &=&8\left( 4\right) =32 \\ \text{RHS} &=&5\left( 2\right) ^{2}+14\left( 2\right) -16=5\cdot 4+14\cdot 2-16=20+28-16 \\ &=&48-16=32 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $% 8y=5x^{2}+14x-16$. \item Is the point $C\left( 4,5\right) $ \ on the graph of \ $% 8y=5x^{2}+14x-16$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a solution of $8y=5x^{2}+14x-16$?% \begin{eqnarray*} \text{LHS} &=&8\left( 5\right) =40 \\ \text{RHS} &=&5\left( 4\right) ^{2}+14\left( 4\right) -16=5\cdot 16+14\cdot 4-16=80+56-16 \\ &=&136-16=120 \end{eqnarray*}% Since LHS$~\not=~$RHS, \ the point $C\left( 4,5\right) $ \ is NOT on the graph of $8y=5x^{2}+14x-16$. \item Is it possible that the graph is that of \ $8y=5x^{2}+14x-16$? \ No, since $C$ is not on the graph of the equation of $8y=5x^{2}+14x-16$. \end{enumerate} \item Consider the equation \ \ $2y-3=x+3$. \begin{enumerate} \item Is the point $A\left( -4,1\right) $ \ on the graph of \ $2y-3=x+3$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=-4$ \ and $y=1$. \ Is this pair a solution of $\ 2y-3=x+3$?% \begin{eqnarray*} \text{LHS} &=&2\left( 1\right) -3=2-3=-1 \\ \text{RHS} &=&-4+3=-1 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of $2y-3=x+3$. \item Is the point $A\left( 2,4\right) $ \ on the graph of \ $2y-3=x+3$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=2$ \ and $y=4$. \ Is this pair a solution of $2y-3=x+3$?% \begin{eqnarray*} \text{LHS} &=&2\left( 4\right) -3=8-3=5 \\ \text{RHS} &=&2+3=5 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $% 2y-3=x+3$. \item Is the point $C\left( 4,5\right) $ \ on the graph of \ $2y-3=x+3$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a solution of $2y-3=x+3$?% \begin{eqnarray*} \text{LHS} &=&2\left( 5\right) -3=10-3=7 \\ \text{RHS} &=&4+3=7 \end{eqnarray*}% Since LHS$~=~$RHS, \ the point $C\left( 4,5\right) $ \ is on the graph of $% 2y-3=x+3$. \item Is it possible that the graph is that of \ $2y-3=x+3$? \ Based on the points $A,$ $B$, and $C$, the answer is yes, since $A,$ $B,$ and $C$ are all on the graph of the equation $2y-3=x+3$. \end{enumerate} \item Consider the equation \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \begin{enumerate} \item Is the point $A\left( -4,1\right) $ \ on the graph of \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=-4$ \ and $y=1$. \ Is this pair a solution of $\ 6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $?% \begin{eqnarray*} \text{LHS} &=&6\left( 1-1\right) =6\left( 0\right) =0 \\ \text{RHS} &=&\left( \left( -4\right) -1\right) \left( \left( -4\right) ^{2}-\left( -4\right) -20\right) =-5\left( 16+4-20\right) =-5\left( 20-20\right) \\ &=&-5\cdot 0=0 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $A\left( -4,1\right) $ \ is on the graph of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \item Is the point $A\left( 2,4\right) $ \ on the graph of \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=2$ \ and $y=4$. \ Is this pair a solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $% ?% \begin{eqnarray*} \text{LHS} &=&6\left( 1-4\right) =6\left( -3\right) =-18 \\ \text{RHS} &=&\left( 2-1\right) \left( 2^{2}-2-20\right) =1\left( 4-2-20\right) =1\left( 2-20\right) \\ &=&1\left( -18\right) =-18 \end{eqnarray*}% Since LHS$~=~$RHS$,$ \ the point $B\left( 2,4\right) $ \ is on the graph of $% 6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \item Is the point $C\left( 4,5\right) $ \ on the graph of \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=4$ \ and $y=5$. \ Is this pair a solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $% ?% \begin{eqnarray*} \text{LHS} &=&6\left( 1-5\right) =6\left( -4\right) =-24 \\ \text{RHS} &=&\left( 4-1\right) \left( 4^{2}-4-20\right) =3\left( 16-4-20\right) =3\left( 12-20\right) \\ &=&3\left( -8\right) =-24 \end{eqnarray*}% Since LHS$~=~$RHS, \ the point $C\left( 4,5\right) $ \ is on the graph of $% 6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \item Is it possible that the graph is that of \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $? \ Based on the points $A,$ $B$, and $C$, the answer is yes, since $A,$ $B,$ and $C$ are all on the graph of the equation $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \end{enumerate} \end{enumerate} We need to rule out one of the equations from b) and c). \ Let us test these equations using point $D\left( -1,-5\right) $. For part b): Is the point $D\left( -1,-5\right) $ \ on the graph of \ $2y-3=x+3$? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=-1$ \ and $y=-5$. \ Is this pair a solution of $2y-3=x+3$?% \begin{eqnarray*} \text{LHS} &=&2\left( -5\right) -3=-10-3=-13 \\ \text{RHS} &=&-1+3=2 \end{eqnarray*}% Since LHS$~\not=~$RHS, \ the point $D\left( -1,-5\right) $ \ is NOT on the graph of $2y-3=x+3$. \ Consequently, \ the graph is NOT that of $2y-3=x+3$. For part c): Is the point $D\left( -1,-5\right) $ \ on the graph of \ $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $? Solution: \ We need to determine whether the coordinates of $A$ are a solution of the equation or not. \ Let $x=-1$ \ and $y=-5$. \ Is this pair a solution of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $% ?% \begin{eqnarray*} \text{LHS} &=&6\left( 1-\left( -5\right) \right) =6\cdot 6=36 \\ \text{RHS} &=&\left( -1-1\right) \left( \left( -1\right) ^{2}-\left( -1\right) -20\right) =-2\left( 1+1-20\right) =-2\left( 2-20\right) \\ &=&-2\left( -18\right) =36 \end{eqnarray*}% Since LHS$~=~$RHS, \ the point $D\left( -1,-5\right) $ \ is on the graph of $% 6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \ Consequently, \ it is possible that the graph is that of $6\left( 1-y\right) =\left( x-1\right) \left( x^{2}-x-20\right) $. \end{enumerate} \end{document}