% %ȉ̕RgƂ̂łC_ł͂܂ł܂B
% ̃^Cg
[Title]
212A_2jifunctionA

% ꂼ̖𓚂$\displaystyle $tꍇ́C@ON ܂1
% ꂼ̖𓚂$\displaystyle $tȂꍇ́COFF܂0
% up̕ҏWv|u[U[ݒv̉ɂݒ
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[displaystyle]
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% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
% ftHǵC5i\normalsizej
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[FontSize]
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% vAuɒǉpbP[Wt@Cw肷B
[usepackage]
\usepackage{color,amsmath,amssymb}
\usepackage{graphicx}

% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
Ot̊T`ibj
}̃Ot2֐̎߂B\\
\textcircled{\small 1}\,$y=x^2+2x+3$\\
\textcircled{\small 2}\,$y=x^2-2x+3$\\
\textcircled{\small 3}\,$y=2x^2+4x+4$\\
\textcircled{\small 4}\,$y=2x^2-4x+4$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic64.tex}
\end{center}

process
_$(1,\,2)$C$y=a(x-1)^2+2$ƂB
ꂪ_$(3,\,6)$ʂ邩C$6=a(3-1)^2+2$C$a=1$B
߂鎮$y=(x-1)^2+2=x^2-2x+3$

\textcircled{\small 2}

̐}́C_$\left( 0C3 \right)$œ_$\left( 1C0 \right)$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic125.tex}
\end{center}   
\textcircled{\small 1}\,$y=3x^2+1$\\
\textcircled{\small 2}\,$y=3x^2-3$\\
\textcircled{\small 3}\,$y=-3x^2+3$\\
\textcircled{\small 4}\,$y=-x^2+3$

process
$y=ax^2+3$$(1C0)$ʂ邩C
$0=a\times 1^2+3$C$\therefore a=-3$

\textcircled{\small 3}

}́C__$(2,-4)$ŁC_ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐\fbox{\bf\, C \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$y=(x+2)^2-4$~~~
\textcircled{\small 2}\,$y=(x-2)^2-4$\\
\textcircled{\small 3}\,$y=2(x+2)^2-4$~~~
\textcircled{\small 4}\,$y=2(x-2)^2-4$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h3001m0301.tex}
\end{center} 

process
_獶$2$̂ƂɁC$4$ł邩C$a=1$

\textcircled{\small 2}

2֐$y=-(x-1)^2+2$̃Ot̊T`ƂāC
łK؂Ȃ̂$\fbox{\bf\,A\,}$łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2701m0301.eps}
 \end{center}\end{figure} 

process
_$(1,\,2)$C
܂C$x=0$̂Ƃ$y=-(0-1)^2+2=1$ł邩
$(0,\,1)$ʂB

\textcircled{\small 2}

2֐$y=2\left( x-3 \right)^2$̃Ot̊T`ƂāCłK؂Ȃ̂͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB\\
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2402M001.eps}
 \end{center}\end{figure} 

process
 _$\left(3,\,0 \right)$łB

\textcircled{\small 3}

2֐$y=2\left(x+1 \right)^2+2$̃Ot̊T`ƂāCłK؂Ȃ̂͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB\\
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2501M001.eps}
 \end{center}\end{figure} 

process
_$\left(-1C2\right)$

\textcircled{\small 2}

2֐$y=\left(x-1 \right)^2-1$̃Ot̊T`ƂāCłK؂Ȃ̂͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂IׁB\\
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2602M001.eps}
 \end{center}\end{figure} 

process
_$\left( 1C-1 \right)$łB

\textcircled{\small 4}

2֐$y=3x^2-1$̃Ot̊T`ƂāCłK؂Ȃ̂\fbox{\bf\, A \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2702m0301.eps}
 \end{center}\end{figure} 

process
_$(0,\,-1)$

\textcircled{\small 4}

2֐$y=x^2+4x+4$̃Ot̊T`ƂčłK؂Ȃ̂
\fbox{\bf\hspace{2pt} A \hspace{2pt}}łB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m301.eps}
 \end{center}\end{figure} 

process
$y=(x+2)^2$ƕό`ł邩C_$(-2,0)$

\textcircled{\small 1}

2֐$y=-2x^2+3$̃Ot̊T`ƂāCłK؂Ȃ̂
\fbox{\bf\, A \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂IׁB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h3002301.eps}
 \end{center}\end{figure} 

\textcircled{\small 4}

2֐$y=(x+2)^2-3$̃Ot̊T`ƂčłK؂Ȃ̂
\fbox{\bf\hspace{4pt} A \hspace{4pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=70mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h31m301.eps}
 \end{center}\end{figure} 

process
_$(-2,-3)$

\textcircled{\small 3}

E̐}́C2֐$y=a(x-p)^2+1$̃OtłB
$a$C$p$̕ɂāCgݍ킹\fbox{\bf\,A\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$a<0Cp>0$~~~~
\textcircled{\small 2}\,$a<0Cp<0$\\
\textcircled{\small 3}\,$a>0Cp>0$~~~~
\textcircled{\small 4}\,$a>0Cp<0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m0301.tex}
\end{center}

process
F$y=2(x+3)^2+1$

\textcircled{\small 4}

}́C_$i1C0j$ŁC_$i0C2j$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic21.tex}
\end{center}
\begin{tabbing}
@@@@@@@@@\=@@@@@@@@@\= \\ 
\textcircled{\small 1}@$y=2(x-1)^2$ \>\textcircled{\small 2}@$y=2(x+1)^2$\\
\textcircled{\small 3}@$y=(x-1)^2$ \>\textcircled{\small 4}@$y=(x+1)^2$
\end{tabbing}

\textcircled{\small 1}

}́C_̍W$\left( 0C-1 \right)$ŁC_$\left( 2C1 \right)$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͎\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic126.tex}
\end{center} 
\textcircled{\small 1}\,$y=x^2-1$\\
\textcircled{\small 2}\,$y=2x^2-1$\\
\textcircled{\small 3}\,$y=\displaystyle\frac{1}{2} x^2-1$\\
\textcircled{\small 4}\,$y=\left(x-2 \right)^2-1$

process
$y=ax^2-1$$\left( 2C1 \right)$ʂ邩C\\
$1=a2^2-1$C$\therefore a=\displaystyle\frac{1}{2}$

\textcircled{\small 3}

}́C__$\left(1C-1 \right)$ŁC_$\left(0C1 \right)$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic123.tex}
\end{center}
\textcircled{\small 1}\,$y=\left( x+1 \right)^2-1$\\
\textcircled{\small 2}\,$y=\left( x-1 \right)^2-1$\\
\textcircled{\small 3}\,$y=2\left( x+1 \right)^2-1$\\
\textcircled{\small 4}\,$y=2\left( x-1 \right)^2-1$

process
$y=a\left(x-1 \right)^2-1$$\left( 0C1 \right)$ʂ邩C\\
$1=a\left( 0-1 \right)^2-1$C$\therefore a=2$

\textcircled{\small 4}

}́C_$(2,\,4)$ŁC_$(0,\,0)$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐̎߂B\\
\textcircled{\small 1}\,$y=-(x-2)^2+4$~~~
\textcircled{\small 2}\,$y=-(x+2)^2+4$\\
\textcircled{\small 3}\,$y=-2(x-2)^2+4$~~~
\textcircled{\small 4}\,$y=-2(x+2)^2+4$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic56.tex}
\end{center}

process
$y=a(x-2)^2+4$_$(0,\,0)$ʂ邩C
$0=a(-2)^2+4$C$a=-1$

\textcircled{\small 1}

}́C_$(1,-4)$ŁC$y$Ƃ̋L__$(0,-2)$ł2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic48.tex}
\end{center}
\begin{tabbing}
@@@@@@@@@@@@\=@@@@@@@@@\= \\ 
\textcircled{\small 1}@$y=2(x+1)^2-4$ \>\textcircled{\small 2}@$y=2(x-1)^2-4$\\
\textcircled{\small 3}@$y=3(x-1)^2+4$ \>\textcircled{\small 4}@$y=3(x-1)^2-4$
\end{tabbing}

\textcircled{\small 2}

__$(2,\,6)$ŁC_$(0,\,-2)$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1} $ y=-(x+2)^2+6$\\
\textcircled{\small 2} $y=-(x-2)^2+6$\\
\textcircled{\small 3} $y=-2(x+2)^2+6$\\
\textcircled{\small 4} $y=-2(x-2)^2+6$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic53.tex}
\end{center}

process
$y=a(x-2)^2+6$_$(0,\,-2)$ʂ邩C
$-2=a(0-2)^2+6$C$a=-2$

\textcircled{\small 4}

}́C_i$-2$C$-4$jŁC_i0C4jʂ
2֐̃OtłBOt̂悤ɂȂ2֐͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic45.tex}
\end{center}
\begin{tabbing}
@@@@@@@@@@@\=@@@@@@@@@@@\= \\ 
\textcircled{\small 1}\,$y=(x+2)^2-4$ \>\textcircled{\small 2}\,$y=(x-2)^2-4$\\
\textcircled{\small 3}\,$y=2(x+2)^2-4$ \>\textcircled{\small 4}\,$y=2(x-2)^2-4$
\end{tabbing}

process
_Eւ̕ψ$2$$y$͏ɕψ$8$C
2֐̊{`$y=2x^2$i$x$$2$$y$$8$jB
āC$y=2(x+2)^2-4$

\textcircled{\small 3}

}́C__$i1,\,2j$ŁC_$i0,\,-1j$ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐͂ǂꂩB\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic23.tex}
\end{center} 
\textcircled{\small 1}@$y=-2(x+1)^2+2$\\
\textcircled{\small 2}@$y=-2(x-1)^2+2$\\
\textcircled{\small 3}@$y=-3(x+1)^2+2$\\
\textcircled{\small 4}@$y=-3(x-1)^2+2$

process
_i1,\,2jŁC$y=-3x^2$𕽍sړ̂C\textcircled{\small 4}

\textcircled{\small 4}

̐}́C_̍W$(0,\,-1)$ŁC_$(2,\,1)$
ʂ2֐̃OtłB
Ot̂悤ɂȂ2֐$\fbox{\bf\,C\,}$
łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=30mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2701m0302.eps}
 \end{center}\end{figure} 

process
$y=ax^2-1$_$(2,\,1)$ʂ邩C\\
$1=a\times 2^2-1~~~\therefore a=\displaystyle\frac{1}{2}$\\
֐$y=\displaystyle\frac{1}{2} x^2-1$

\textcircled{\small 3}






[Level2]
2֐̒_ibj
2֐$y=-2(x-2)^2+3$̒_̍W߂B

_$(2C\,3)$

2֐$y=5(x+2)^2-3$̒_̍W߂B

_$(-2C\,-3)$

2֐$y=-2(x+3)^2+2$̒_̍W߂B

$(-3C2)$

2֐$y=x^2+10x+5$̃Ot̒_̍W߂B

process
$y=(x+5)^2-5^2+5=(x+5)^2-20$\\
Ē_$(-5C-20)$\\
iʉj$x=\displaystyle\frac{-10}{2\times 1}=-5$\\
_$y$W$=(-5)^2+10\times (-5)+5=-20$\\
iʉj$x^2+10x$܂łƁC$y=x(x+10)$B
̊֐$x$$0$$-10$Ō邩,_$x$W$-5$B\\
_$y$W,$(-5)(-5+10)+5=-20$

$(-5C-20)$

}́C2֐$y=2x^2-4x$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic24.tex}
\end{center}

process
_$x$W$\displaystyle \frac{0+2}{2}=1$B
_$y$W\\
$y=2 \times 1^2-4 \times 1=2-4=-2$

$i1C\,-2j$

2֐$y=(x+1)^2+2$̒_̍W߂B

$(-1C\, 2)$

2֐$y=2(x-3)^2-4$̒_̍W߂B

$(3C\, -4)$

2֐$y=(x-3)^2+2$̒_߂B

$(3,\,2)$

2֐$y=-(x+1)^2+2$̒_̍W߂B

$(-1C\,2)$

2֐$y=-3(x+2)^2+1$̃Ot̒_߂B

$(-2, 1)$

2֐$y=2x^2-3$̃Ot̒_߂B

$(0C\,-3)$

2֐$y=4(x-3)^2-2$̒_̍W߂B

$(3C\,-2)$

2֐$y=2(x+1)^2+3$̃Ot̒_̍W߂B

$(-1,\,3)$

}́C2֐$y=-x^2+6x+7$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic52.tex}
\end{center}

process
i@1j\\
$y=-x^2+6x+7=-(x-3)^2+3^2+7\\
=-(x-3)^2+16$\\
i@2j
Ot蒸_$x$W$\displaystyle \frac{-1+7}{2}=3$B\\
āC_$y$W\\
$y=-3^2+6 \times 3+7=16$

$(3C\, 16)$

2֐$y=x^2+6x+3$̃Ot̒_̍W߂B

process
$y=x^2+6x+3=(x+3)^2-3^2+3=(x+3)^2-6$\\
iʉj$x=\displaystyle\frac{-6}{2\times 1}=-3$\\
_$y$W$=(-3)^2+6\times (-3)+3=-6$\\
iʉj$x^2+6x$܂łƁC$y=x(x+6)$B
̊֐$x$$0$$-6$Ō邩,_$x$W$-3$B\\
_$y$W,$(-3)(-3+6)+3=-6$

$(-3C\,-6)$

}́C2֐$y=-x^2+4x+5$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic42.tex}
\end{center}

process
i@1j\\
$x$W$-1$$5$̒_C$\displaystyle \frac{(-1)+5}{2}=2$\\
$y=-2^2+4 \times 2 +5=9$ \\
i@2j\\
$y=-(x^2-4x)+5=-(x-2)^2+2^2+5\\ =-(x-2)^2+9$C\\
_$i2C9j$

$i2C9j$

2֐$y=x^2+2x+11$̃Ot̒_̍W߂B

process
$y=(x+1)^2-1+11=(x+1)^2+10$\\
iʉj$x=\displaystyle\frac{-2}{2\times 1}=-1$\\
_$y$W$=(-1)^2+2\times (-1)+11=10$\\
iʉj$x^2+2x$܂łƁC$y=x(x+2)$B
̊֐$x$$0$$-2$Ō邩,_$x$W$-1$B\\
_$y$W,$(-1)(-1+2)+11=10$

$i-1C\,10j$

}́C2֐$y=-x^2+6x$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic49.tex}
\end{center}

process
}C_$x$W$3$B
_$y$ẂC$y=-3^2+6 \times 3=9$

$(3C\,9)$

2֐$y=x^2-2x+3$̃Ot̒_̍W߂B

process
$y=(x-1)^2-1+3=(x-1)^2+2$\\
iʉj$x=\displaystyle\frac{-(-2)}{2\times 1}=1$\\
_$y$W$=1^2-2\times 1+3=2$\\
iʉj$x^2-2x$܂łƁC$y=x(x-2)$B
̊֐$x$$0$$2$Ō邩,_$x$W$1$B\\
_$y$W,$1\times (1-2)+3=2$

$(1C\,2)$

}́C2֐$y=2(x+1)(x-3)$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic57.tex}
\end{center}

process
_$x$ẂC
$\displaystyle\frac{-1+3}{2}=1$C
_$y$ẂC
$y=2(1+1)(1-3)=-8$

$(1C\,-8)$

2֐$y=x^2+4x+1$̃Ot̒_̍W߂B

process
$y=(x+2)^2-4+1=(x+2)^2-3$\\
iʉj$x=\displaystyle\frac{-4}{2\times 1}=-2$\\
_$y$W$=(-2)^2+4\times (-2)+1=-3$\\
iʉj$x^2+4x$܂łƁC$y=x(x+4)$B
̊֐$x$$0$$-4$Ō邩,_$x$W$-2$B\\
_$y$W,$(-2)(-2+4)+1=-3$

$(-2C\,-3)$

2֐$y=x^2-4x$̃Ot̒_̍W߂B

process
$y=(x-2)^2-2^2=(x-2)^2-4$\\
Ē_$(2C-4)$\\
iʉj$x=\displaystyle\frac{-(-4)}{2\times 1}=2$\\
_$y$W$=2^2-4\times 2=-4$\\
iʉj$x^2-4x$܂łƁC$y=x(x-4)$B
̊֐$x$$0$$4$Ō邩,_$x$W$2$B\\
_$y$W,$2 \times (2-4)=-4$

$(2C-4)$

2֐$y=x^2+8x+17$̃Ot̒_̍W߂B

process
$x^2+8x$܂łƁC$y=x(x+8)$B
̊֐$x$$0$$-8$Ō邩,_$x$W$-4$B\\
_$y$W,$-4(-4+8)+17=1$

$(-4C1)$

2֐$y=x^2-6x+5$̃Ot̒_̍W߂B

process
$y=(x-3)^2-3^2+5=(x-3)^2-4$\\
Ē_$(3C-4)$\\
iʉj$x=\displaystyle\frac{-(-6)}{2\times 1}=3$\\
_$y$W$=3^2-6\times 3+5=-4$\\
iʉj$x^2-6x$܂łƁC$y=x(x-6)$B
̊֐$x$$0$$6$Ō邩,_$x$W$3$B\\
_$y$W,$3\times (3-6)+5=-4$

$(3C-4)$

2֐$y=x^2-8x$̃Ot̒_̍W߂B

process
$y=(x-4)^2-4^2=(x-4)^2-16$\\
Ē_$(4C-16)$\\
iʉj$x=\displaystyle\frac{-(-8)}{2\times 1}=4$\\
_$y$W$=4^2-8\times 4=-16$\\
iʉj$x^2-8x$܂łƁC$y=x(x-8)$B
̊֐$x$$0$$8$Ō邩,_$x$W$4$B\\
_$y$W,$4 (4-8)=-16$

$(4C-16)$

2֐$y=x^2-4x+3$̃Ot̒_̍W߂B

process
$y=(x-2)^2-2^2+3=(x-2)^2-1$\\
Ē_$(2C-1)$\\
iʉj$x=\displaystyle\frac{-(-4)}{2\times 1}=2$\\
_$y$W$=2^2-4\times 2+3=-1$\\
iʉj$x^2-4x$܂łƁC$y=x(x-4)$B
̊֐$x$$0$$4$Ō邩,_$x$W$2$B\\
_$y$W,$2(2-4)+3=-1$

$(2C-1)$

2֐$y=x^2-10x$̃Ot̒_̍W߂B

process
$y=(x-5)^2-5^2=(x-5)^2-25$\\
Ē_$(5C-25)$\\
iʉj$x=\displaystyle\frac{-(-10)}{2\times 1}=5$\\
_$y$W$=5^2-10\times 5=-25$\\
iʉj$x^2-10x$܂łƁC$y=x(x-10)$B
̊֐$x$$0$$10$Ō邩,_$x$W$5$B\\
_$y$W,$5 \times (5-10)=-25$

$(5C-25)$

2֐$y=x^2+8x+10$̃Ot̒_̍W߂B

process
$y=(x+4)^2-4^2+10=(x+4)^2-6$\\
Ē_$(-4C-6)$\\
iʉj$x=\displaystyle\frac{-8}{2\times 1}=-4$\\
_$y$W$=(-4)^2+8\times (-4)+10=-6$\\
iʉj$x^2+8x$܂łƁC$y=x(x+8)$B
̊֐$x$$0$$-8$Ō邩,_$x$W$-4$B\\
_$y$W,$(-4)(-4+8)+10=-6$

$(-4C-6)$

\hspace{8pt} ̒̕_̍W߂B\\
$y=-3\left( x+\displaystyle\frac{2}{3} \right)^2+\displaystyle\frac{7}{3}$

$\left( -\displaystyle\frac{2}{3} C\,\displaystyle\frac{7}{3}\right)$

\hspace{8pt} ̎̕̕߂B\\
$y=x^2-3x+4$

process
$y=x^2-3x$́C$y=x(x-3)$ƈC$x$03Ō邩C$x=\displaystyle\frac{3}{2}$B\\
iʉj
$y=\left( x-\displaystyle\frac{3}{2} \right)^2\cdots$vׂ

$x=\displaystyle\frac{3}{2}$

\hspace{8pt} ̒̕_̍W߂B\\
$y=x(x-2)-3$

process
$y=x(x-2)$$x$$0$2Ō邩C$x=1$B\\
āC_$y$W\\
$y=1\times (1-2)-3=-4$

$(1C-4)$

2֐$y=x^2-4x-5$̃Ot̒_̍W߂B

process
$x^2-4x$܂łƁC$y=x(x-4)$B
̊֐$x$$0$$4$Ō邩,_$x$W$2$B\\
_$y$W,$2(2-4)-5=-9$

$(2C-9)$

2֐$y=x^2+6x+5$̃Ot̒_̍W߂B

process
$x^2+6x$܂łƁC$y=x(x+6)$B
̊֐$x$$0$$-6$Ō邩,_$x$W$-3$B\\
_$y$W,$-3(-3+6)+5=-4$

$(-3C-4)$

2֐$y=x^2-2x-5$̃Ot̒_̍W߂B

process
$y=(x-1)^2-1^2-5=(x-1)^2-6$\\
Ē_$(1C-6)$\\
iʉj$x^2-2x$܂łƁC$y=x(x-2)$B
̊֐$x$$0$$2$Ō邩,_$x$W$1$B\\
_$y$W,$1^2-2\times 1-5=-6$

$(1C-6)$

\hspace{8pt} ̒̕_̍W߂B\\
$y=2(x-3)(x-7)+1$

process
$y=2(x-3)(x-7)$$x$37Ō邩C$x=\displaystyle\frac{3+7}{2}=5$\\
_$y$W$y=2(5-3)(5-7)=-8$

$i5C-8j$

̐}́C2֐$y=-\left(x-2 \right) \left(x+4 \right)$̃OtłB
̃Ot̒_߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic121.tex}
\end{center}

process
_$x$W$-4$$2$̒_łB\\
$\displaystyle\frac{-4+2}{2}=-1$\\
_$y$W\\
$y=-\left(-1-2 \right) \left(-1+4 \right)=9$

$\left(-1,\,9 \right)$

2֐$y=x^2+8x$̃Ot̒_̍W߂B

process
$y=x(x+8)$ƈłC2֐$x$$0$C$-8$Ō邩C
_$x$W$-4$B\\
_$y$W$y=(-4) (-4+8)=-16$\\
iʉj$x^2+8x$܂łƁC$y=x(x+8)$B
̊֐$x$$0$$-8$Ō邩,_$x$W$-4$B\\
_$y$W,$(-4)(-4+8)=-16$

$(-4C-16)$

̐}́C2֐$y=-2x^2+4x$̃OtłB
̃Ot̒_̍W߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic124.tex}
\end{center}

process
_$x$W$\displaystyle\frac{0+2}{2}=1$B
$y$W$y=-2\times 1^2+4\times 1=2$

$\left( 1C2 \right)$

2֐$y=x^2-4x+8$̃Ot̒_̍W߂B

process
$y=x^2-4x$܂łƁC$y=x(x-4)$ƂȂ邩C
$x$$0$$4$ŌBāC$x$W$2$B
$y$W$y=2^2-4 \times 2+8=4$\\
iʉj$x^2-4x$܂łƁC$y=x(x-4)$B
̊֐$x$$0$$4$Ō邩,_$x$W$2$B\\
_$y$W,$2\times (2-4)+8=4$

$\left( 2C4 \right)$

2֐$y=x^2-6x$̃Ot̒_̍W߂B

process
$y=x^2-6x$܂łƁC$y=x(x-6)$ƂȂ邩C
$x$$0$$6$ŌBāC_$x$W$3$B
_$y$W$y=3\times (3-6)=-9$

$\left( 3C-9 \right)$

̐}́C2֐$y=x^2+2x+1$̃OtłB
̃Ot̒_̍W$\left(\fbox{\bf EG},\,0\right)$łB
\begin{figure}[hbt]\begin{center}
 \includegraphics[width=30mm,clip]{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2702m0302.eps}
 \end{center}\end{figure} 

process
$y=(x+1)^2$~~~
_$(-1,\,0)$

$-1$

̐}́C2֐$y=-\left(x+3 \right) \left(x-1 \right)$̃OtłB
̃Ot̒_߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic122.tex}
\end{center}

process
_$x$W$-3$$1$̒_łB\\
$\displaystyle\frac{-3+1}{2}=-1$\\
_$y$W\\
$y=-\left(-1+3 \right) \left(-1-1 \right)=4$

$\left(-1,\,4 \right)$

}́C2֐$y=-x^2+2x$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m0302.tex}
\end{center}

process
_$x$ẂC
$\displaystyle\frac{0+2}{2}=1$C
_$y$ẂC
$y=-1^2+2\times 1=1$

$(1C\,1)$

$y=x^2+ax-b$̒_$(-2,\,3)$łƂC$a$C$b$̒l߂\fbox{\hspace{8pt} 8 \hspace{8pt}}łB\\
\textcircled{\small 1}\,
$\left\{ 
\begin{array}{l}
a=-4 \\
b=-7
\end{array} \right.$~
\textcircled{\small 2}\,
$\left\{ 
\begin{array}{l}
a=-4 \\
b=7
\end{array} \right.$\\
\textcircled{\small 3}\,
$\left\{ 
\begin{array}{l}
a=4 \\
b=-7
\end{array} \right.$~
\textcircled{\small 4}\,
$\left\{ 
\begin{array}{l}
a=4 \\
b=7
\end{array} \right.$

process
$y=(x+2)^2+3=x^2+4x+7$

\textcircled{\small 3}

}́C2֐$y=2x^2+4x$̃OtłB
̃Ot̒_̍W$(\fbox{\bf EG}, \fbox{\bf IJ})$łB\\
\textcircled{\small 1}\,$(-1,\,-2)$~~
\textcircled{\small 2}\,$(-1,\,-3)$~~
\textcircled{\small 3}\,$(-1,\,-4)$~~
\textcircled{\small 4}\,$(-1,\,-5)$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2902m0303.tex}
\end{center}

\textcircled{\small 1}

}́C2֐$y=-x^2+6x-5$̃OtłB
̃Ot̒_̍Ẃi\fbox{\bf\, E \,}C\fbox{\bf\, G \,}jłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m302.tex}
\end{center}

$(3,\,4)$

}2֐$y=-x^2+2x+8$̃OtłB
̃Ot̒_̍W
\textcircled{\small 1}\,$(1,\,8)$~~
\textcircled{\small 2}\,$(1,\,9)$~~
\textcircled{\small 3}\,$(2,\,8)$~~
\textcircled{\small 4}\,$(2,\,9)$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2802m0302.tex}
\end{center}

\textcircled{\small 2}

2֐̒_߂B\\
$y=x^2-4x+5$

process
$y=(x-2)^2-2^2+5=(x-2)^2+1$

$(2,\,1)$

2֐$y=-3x^2+6x$̃Ot̒_̍W߂B
C}2֐$y=-3x^2+6x$̃OtłB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h3001m0302.tex}
\end{center} 

$(1,\,3)$

2֐$y=x^2+4x+6$̃Ot̒_̍W߂B

$(-2,\,2)$

2֐$y=x^2+6x+9+k$i$k$͒萔j̃Ot
̒_$y$W$4$łƂC$k$̒l
\fbox{\bf\hspace{4pt} E \hspace{4pt}}łB\\
\textcircled{\small 1}\,$1$~~~
\textcircled{\small 2}\,$2$~~~
\textcircled{\small 3}\,$3$~~~
\textcircled{\small 4}\,$4$

process
$y=x(x+6)+9+k$ƕό`ł邩C$x=-3$\\
Ē_$(-3,4)$ƂȂ͂łB\\
$4=(-3)^2+6(-3)+9-k$~~~$\therefore k=4$

\textcircled{\small 4}

2֐$y=x^2-2x-3$
Ot̒_̍WƂĐ̂C
\textcircled{\small 1}$\sim$
\textcircled{\small 6}̒1IсC
̔ԍ𓚂ȂB\\
\textcircled{\small 1}\,$(-1,-4)$~~~
\textcircled{\small 2}\,$(1,-4)$~~~
\textcircled{\small 3}\,$(-1,4)$\\
\textcircled{\small 4}\,$(1,4)$~~~
\textcircled{\small 5}\,$(1,-3)$~~~
\textcircled{\small 6}\,$(2,-4)$

process
$y=x^2-2x-3=x(x-2)-3$ƕό`ł邩C$x=1$

\textcircled{\small 2}

2֐$y=x^2-2x-3$̎$y=a(x-p)^2+q$̌`ɕό`ƂɁC
͂ǂꂩB\textcircled{\small 1}$\sim$
\textcircled{\small 6}̒1IсC
̔ԍ𓚂ȂB\\
\textcircled{\small 1}\,$y=(x-1)^2-3$~~~
\textcircled{\small 2}\,$y=(x-1)^2-2$\\
\textcircled{\small 3}\,$y=(x-1)^2-4$~~~
\textcircled{\small 4}\,$y=(x-2)^2-3$\\
\textcircled{\small 5}\,$y=(x+1)^2-3$~~~
\textcircled{\small 6}\,$y=(x+1)^2-4$

process
$y=x^2-2x-3=(x-1)^2-1-3\\ =(x-1)^2-4$

\textcircled{\small 3}







[Level3]
sړibj
2֐$y=-\left( x-3 \right)^2+7$̃Ot́C
2֐$y=-x^2$̃Otǂ̂悤ɕsړ̂B
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$x$3C$y$$7$sړ\\
\textcircled{\small 2}\,$x$3C$y$$-7$sړ\\
\textcircled{\small 3}\,$x$$-3$C$y$$-7$sړ\\
\textcircled{\small 4}\,$x$$-3$C$y$$7$sړ

process
$y=-x^2$̒_͌_łB
$y=-\left( x-3 \right)^2+7$̒_$\left(3C7 \right)$łB

\textcircled{\small 1}

2֐$y=2x^2$̃Ot$x$$-5$C$y$$-3$sړB
̂ƂCړ̋ȐOtƂ2֐͂ǂꂩB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$y=2\left( x-5 \right)^2-3$\\
\textcircled{\small 2}\,$y=2\left( x-5 \right)^2+3$\\
\textcircled{\small 3}\,$y=2\left( x+5 \right)^2-3$\\
\textcircled{\small 4}\,$y=2\left( x+5 \right)^2+3$

process
$y=2\{ x-(-5) \}-3$

\textcircled{\small 3}

2֐$y=x^2-2x-3$
Ot$x$$-2$C$y$$3$sړƂC̃Ot́Cǂ̂悤Ȏŕ\邩
̂C
\textcircled{\small 1}$\sim$
\textcircled{\small 6}̒1IсC
̔ԍ𓚂ȂB\\
\textcircled{\small 1}\,$y=x^2-2x$~~~
\textcircled{\small 2}\,$y=x^2-6x+8$\\
\textcircled{\small 3}\,$y=x^2-6x+5$~~~
\textcircled{\small 4}\,$y=x^2+2x$\\
\textcircled{\small 5}\,$y=x^2+2x+3$~~~
\textcircled{\small 6}\,$y=x^2-2x-2$

process
$y-3=(x+2)^2-2(x+2)-3$~~~
$\therefore y=x^2+2x$

\textcircled{\small 4}

2֐$y=-2x^2$̃Ot$x$$p$C$y$$-4$sړƁC
$y=-2(x-2)^2+q$̃OtꂽB̂ƂC$p$C$q$̒lƂĐgݍ킹\fbox{\hspace{2pt} {\bf A} }łB\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB\\
\textcircled{\small 1}\,$p=2Cq=4$~~~
\textcircled{\small 2}\,$p=-2Cq=-4$\\
\textcircled{\small 3}\,$p=-2Cq=4$~~~
\textcircled{\small 4}\,$p=2Cq=-4$

\textcircled{\small 4}

2֐$y=x^2$̃Ot$y$2sړB
̂ƂCړ̋ȐOtƂ2֐\fbox{\bf\hspace{2pt} A \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$y=(x+2)^2$~~~
\textcircled{\small 2}\,$y=(x-2)^2$\\
\textcircled{\small 3}\,$y=x^2-2$~~~
\textcircled{\small 4}\,$y=x^2+2$

\textcircled{\small 4}

2֐$y=x^2$̃Ot$x$2sړB
̂ƂCړ̋ȐOtƂ2֐\fbox{\bf\hspace{2pt} A \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$y=(x+2)^2$~~~
\textcircled{\small 2}\,$y=(x-2)^2$\\
\textcircled{\small 3}\,$y=x^2-2$~~~
\textcircled{\small 4}\,$y=x^2+2$

\textcircled{\small 2}

2֐$y=2x^2$̃Ot$x$$-1$C$y$$3$
sړB̂ƂCړ̋ȐOtƂ2֐
\fbox{\bf\, A \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,$y=2(x-1)^2-3$~~~
\textcircled{\small 2}\,$y=2(x-1)^2+3$\\
\textcircled{\small 3}\,$y=2(x+1)^2-3$~~~
\textcircled{\small 4}\,$y=2(x+1)^2+3$

\textcircled{\small 4}

$y=x^2$̃Ot$x$$3$sړ߂B

process
͂߂̒_i0C\,0jCړi3C\,0jC̒_i3C\,0j

$y=(x-3)^2=x^2-6x+9$

$y=-x^2$̃Ot$y$$5$sړ߂B

process
͂߂̒_i0,\,0jCړi0,\,5jC̒_i0,\,5j

$y=-x^2+5$

$y=3x^2$̃Ot$x$$-2$C$y$$-4$sړ߂B

process
͂߂̒_i0,\,0jCړ$i-2,\,-4j$C̒_$i-2,\,-4j$\\
$y=3(x+2)^2-4=3(x^2+4x+4)-4=3x^2+12x+8$

$y=3(x+2)^2-4=3x^2+12x+8$

2֐$y=2x^2$̃Ot$x$$4$C$y$$-3$sړB
̂ƂCړ̋ȐOtƂ2֐͂ǂꂩB
\textcircled{\small 1}`\textcircled{\small 4}̂琳̂IׁB\\
\begin{tabbing}
@@@@@@@@@@@\=@@@@@@@@@\= \\ 
\textcircled{\small 1} $y=2(x-4)^2+3$ \>\textcircled{\small 2} $y=2(x-4)^2-3$\\
\textcircled{\small 3} $y=2(x+4)^2+3$ \>\textcircled{\small 4} $y=2(x+4)^2-3$
\end{tabbing}

process
_͐}̂悤$i4,-3j$COt\\
$y=2(x-4)^2-3$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic41.tex}
\end{center}

\textcircled{\small 2}

$y=x^2$̒_$(2,\,1)$ɂȂ悤ɕsړƂ2֐̎߂B

process
͂߂̒_i0,\,0jCړ$i2,\,1j$C̒_$i2,\,1j$

$y=(x-2)^2+1=x^2-4x+5$

$y=-2x^2$$x$$-2$C$y$3sړƂ2֐̎߂B

process
͂߂̒_i0,\,0jCړ$i-2,\,3j$C̒_$i-2,\,3j$

$y=-2(x+2)^2+3=-2x^2-8x-5$
















[Level4]
2֐̎ia>=2,-1Cbj
\hspace{8pt} ̎̕̕߂B\\
$y=-x^2-4x+5$

process
$y=-x^2-4x$́C$y=-x(x+4)$ƈC$x$0$-4$Ō邩C$x=-2$B\\
iʉj$y=-(x+2)^2\cdots$vׂ

$x=-2$

\hspace{8pt} ̎̕̕߂B\\
$y=-3x^2+3x+1$

process
$y=-3x^2+3x$́C$y=-3x(x-1)$ƈC$x$0$1$Ō邩C$x=\displaystyle\frac{1}{2}$B\\
iʉj$y=-3\left(x-\displaystyle\frac{1}{2} \right)^2\cdots$vׂ

$x=\displaystyle\frac{1}{2}$

\hspace{8pt} ̎̕̕߂B\\
$y=\displaystyle\frac{1}{2}x^2+11x+9$

process
$y=\displaystyle\frac{1}{2}x^2+11x$́C$y=\displaystyle\frac{1}{2}x(x+22)$ƈC$x$0$-22$Ō邩C$x=-11$B\\
iʉj$y=\displaystyle\frac{1}{2}\left(x+11\right)^2\cdots$vׂ

$x=-11$

\hspace{8pt} ̎̕̕߂B\\
$y=-3x^2+(\sqrt{3}+1)x+1$

process
^$=-3\left( x^2-\displaystyle\frac{\sqrt{3}+1}{3}x \right)+1\\
=-3\left( x-\displaystyle\frac{\sqrt{3}+1}{6} \right)^2+\cdots$vׂ

$x=\displaystyle\frac{\sqrt{3}+1}{6}$

\hspace{8pt} ̎̕̕߂B\\
$y=ax^2+(a+1)x+a^2-1$

process
^$=a\left( x^2+\displaystyle\frac{a+1}{a}x \right)+a^2-1\\
=a\left( x+\displaystyle\frac{a+1}{2a}\right)^2+\cdots$vׂ\\
$x=-\displaystyle\frac{a+1}{2a}$

$x=-\displaystyle\frac{a+1}{2a}$

\hspace{8pt} ̎̕̕߂B\\
$y=\pi x^2+(1-2\pi)x+1-\pi^2$

process
$y=\pi \left( x^2+\displaystyle\frac{1-2\pi}{\pi}\right)+\cdots\\
=\pi \left( x+\displaystyle\frac{1-2\pi}{2\pi}\right)^2\cdots$vׂ\\
$x=-\displaystyle\frac{1-2\pi}{2\pi}$

$x=-\displaystyle\frac{1-2\pi}{2\pi}=1-\displaystyle\frac{1}{2\pi}$

\hspace{8pt} ̎̕̕߂B\\
$y=3x^2+7x-8$

process
$y=3\left(x^2+\displaystyle\frac{7}{3}x \right)+\cdots\\
=3\left(x+\displaystyle\frac{7}{6} \right)+\cdots$

$x=-\displaystyle\frac{7}{6}$





[Level5]
2֐̒_Csړia>=2,-1bj
$y=-2x^2+4$̃Ot$x$$-4$C$y$$3$sړ߂B

process
͂߂̒_i0,\,4jCړ$i-4,\,3j$C̒_$i-4,\,7j$\\
$y=-2(x+4)^2+7=-2(x^2+8x+16)+7=-2x^2-16x-25$

$y=-2(x+4)^2+7=-2x^2-16x-25$

$y=-x^2+4x+3$̃Ot$x$$1$C$y$$-2$sړ߂B

process
$y=-(x^2-4x)+3=-(x-2)^2+4+3=-(x-2)^2+7$̃Ot$x$$1$C$y$$-2$sړƁC\\
͂߂̒_i2,\,7jCړ$i1,\,-2j$C̒_$i3,\,5j$\\
$y=-(x-3)^2+5=-x^2+6x-4$\\
iʉj$y=-(x-1)^2+4(x-1)+3-2$

$y=-(x-3)^2+5=-x^2+6x-4$

2֐$y=2x^2-8x-5$̃Ot̒_̍W߂B

process
$y=2(x^2-4x)-5=2(x-2)^2-2\cdot 2^2-5=2(x-2)^2-13$\\
Ē_$(2,-13)$\\
iʉj$x=\displaystyle\frac{-(-8)}{2\times 2}=2$\\
_$y$W$=2^2-8\times 2-5=-13$\\

$(2,-13)$

2֐$y=-x^2-6x-4$̃Ot̒_̍W߂B

process
$y=-(x^2+6x)-4=-(x+3)^2+3^2-4=-(x+3)^2+5$\\
Ē_$(-3,5)$\\
iʉj$x=\displaystyle\frac{-(-6)}{2\times (-1)}=-3$\\
_$y$W$=-(-3)^2-6\times (-3)-4=5$\\

$(-3,5)$

2֐$y=2x^2-4x+3$̃Ot̒_̍W߂B

process
$y=2(x^2-2x)+3=2(x-1)^2-2\cdot 1^2+3=2(x-1)^2+1$\\
Ē_$(1,1)$\\
iʉj$x=\displaystyle\frac{-(-4)}{2\times 2}=1$\\
_$y$W$=2\times 1^2-4\times 1+3=1$\\

$(1,1)$

2֐$y=-x^2-4x-1$̃Ot̒_̍W߂B

process
$y=-(x^2+4x)-1=-(x+2)^2+2^2-1=-(x+2)^2+3$\\
Ē_$(-2,3)$\\
iʉj$x=\displaystyle\frac{-(-4)}{2\times (-1)}=-2$\\
_$y$W$=-(-2)^2-4\times (-2)-1=3$\\

$(-2,3)$

2֐$y=3x^2+6x+5$̃Ot̒_̍W߂B

process
$y=3(x^2+2x)+5=3(x+1)^2-3\cdot 1^2+5=3(x+1)^2+2$\\
Ē_$(-1,2)$\\
iʉj$x=\displaystyle\frac{-6}{2\times 3}=-1$\\
_$y$W$=3\times (-1)^2+6\times (-1)+5=2$\\

$(-1,2)$

2֐$y=-x^2+2x-4$̃Ot̒_̍W߂B

process
$y=-(x^2-2x)-4=-(x-1)^2+1^2-4=-(x-1)^2-3$\\
Ē_$(1,-3)$\\
iʉj$x=\displaystyle\frac{-2}{2\times (-1)}=1$\\
_$y$W$=-1^2+2\times 1-4=-3$\\

$(1,-3)$

}́C2֐$y=-(x+2)(x-4)$̃OtłB
̃Ot̒_̍W߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic22.tex}
\end{center}

process
_$x$W$\displaystyle \frac{-2+4}{2}=1$C
_$y$W$-(1+2)(1-4)=-3 \times (-3)=9$

(1,\,9)

$y=\displaystyle\frac{1}{2}x^2-2x+3$̒_߂B

process
$y=\displaystyle\frac{1}{2}x^2-2x+3=\displaystyle\frac{1}{2}(x^2-4x)+3\\
=\displaystyle\frac{1}{2}(x-2)^2-\displaystyle\frac{1}{2}\times 2^2+3\\
=\displaystyle\frac{1}{2}(x-2)^2+1$

$(2C1)$

$y=2x^2-8x+3$̒_߂B

process
$y=2(x^2-4x)+3=2(x-2)^2-2\times 2^2+3\\
=2(x-2)^2-5$

$(2C-5)$

\hspace{8pt} ̒̕_̍W߂B\\
$y=-\displaystyle\frac{1}{3}(x-1)(x-3)$

process
^$x$13Ō邩C$x=2$B\\
āC_$y$W$y=-\displaystyle\frac{1}{3}(2-1)(2-3)=\displaystyle\frac{1}{3}$

$\left( 2C\,\displaystyle\frac{1}{3} \right)$

\hspace{8pt} ̒̕_̍W߂B\\
$y=(x+4)(x-1)$

process
^$x$$-4$1Ō邩C$x=\displaystyle\frac{-4+1}{2}=-\displaystyle\frac{3}{2}$B\\
āC_$y$W\\
$y=\left(-\displaystyle\frac{3}{2}+4 \right)\left(-\displaystyle\frac{3}{2}-1\right)=-\displaystyle\frac{25}{4}$

$\left( -\displaystyle\frac{3}{2}C\,-\displaystyle\frac{25}{4} \right)$







[Level6]
2֐̑Ώ̈ړibj
$y=x^2+6x+5$
_ɊւđΏ̈ړOtƂ2֐߂ȂB

process
$(x,y)$_ɊւđΏ̈ړ_$(X,Y)=(-x, -y)$ł邩C
$x=-X, y=-Y$ƂȂBāC߂\\
$-Y=(-X)^2+6(-X)+5$\\
$\therefore -Y=x^2-6X+5$~~~
$Y=-X^2+6X-5$

$y=-x^2+6x-5$

$y=-x^2+2x+3$̃Ot$x$ɂđΏ̈ړOt̎߂B

process
$y$$-y$ƒuƁC$-y=-x^2+2x+3$C$y=x^2-2x-3$

$y=x^2-2x-3$

$y=-x^2+2x+3$̃Ot$y$ɂđΏ̈ړOt̎߂B

process
$x$$-x$ƒuƁC$y=-(-x)^2+2(-x)+3$C$y=x^2-2x+3$

$y=x^2-2x+3$

$y=-x^2+2x+3$̃Ot_ɂđΏ̈ړOt̎߂B

process
$x$$-x$C$y$$-y$ƒuƁC$-y=-(-x)^2+2(-x)+3$C$y=-x^2+2x-3$

$y=-x^2+2x-3$

$y=-x^2+2x+3$̃Ot$y=x$ɂđΏ̈ړOt̎߂B

process
$x$$y$ɁC$y$$x$ɒuƁC$x=-y^2+2y+3$

$x=-y^2+2y+3$

$y=x^2+8x+17$$y$ɊւđΏ̈ړOtƂ
2֐߂B

process
$y=(-x)^2+8(-x)+17$C$y=x^2-8x+17$

$y=x^2-8x+17$











[Level7]
ʂ_ibj
2֐$y=2(x-1)^2-k$i$k$͒萔j̃Ot_$(3,\,5)$ʂƂC
$k$̒l\fbox{\bf\, C \,}łB

process
$5=2(3-1)^2-k$$k=3$

3

2֐$y=x^2+kx-2$i$k$͒萔j̃Ot_$(-3,\,1)$ʂƂC
$k$̒l߂B

process
$1=(-3)^2+k\times(-3)-2$$k=2$

2

2֐$y=a\left(x-2\right)^2+1$i$a$͒萔j̃Ot
_$\left(3C0 \right)$ʂƂC$a$̒l߂B

process
$0=a(3-2)^2+1$C$\therefore a=-1$

$-1$

2֐$y=a(x-2)^2-4$i$a$͒萔j̃Ot_$(3,\,0)$ʂƂC
$a$̒l$\fbox{\bf\,C\,}$łB

process
$0=a(3-2)^2-4$~~~
$0=a-4$

$4$

2֐$y=-\left( x-2 \right)^2+k$i$k$͒萔j̃Ot_ʂƂC
$k$̒l߂B

process
$0=-\left( 0-2 \right)^2+k$C$k=4$

$4$

Q֐̒_̍W$i1C-2j$łC_$i0C-1j$ʂƂĈQ֐̕߂B

process
$y=a(x-1)^2-2$_$i0C-1j$ʂ邩$-1=a(0-1)^2-2$\hspace{8pt}$\therefore a=1$\\
$y=(x-1)^2-2=x^2-2x-1$

$y=(x-1)^2-2=x^2-2x-1$

Q֐$x$Ɠ_$i0C0j$C$i2C0j$ŌC_$i-1C3j$ʂƂĈQ֐̕߂B

process
$y=ax(x-2)$_$i-1C3j$ʂ邩C
$3=a(-1)(-1-2)$\hspace{8pt}$\therefore a=1$\\
$\therefore y=x(x-2)=x^2-2x$

$y=x^2-2x$

2֐$y=a\left(x+2\right)^2-3$i$a$͒萔j̃Ot
_$\left(-4C5 \right)$ʂƂC$a$̒l߂B

process
$5=a(-4+2)^2-3$C$\therefore a=2$

2

2֐$y=2(x-3)^2+k$ik͒萔j̃Ot_(0,\,13)ʂƂC
$k$̒l߂B

process
$13=2(0-3)^2+k$C$13=2 \times 9 +k$C\\
$\therefore k=13-18=-5$

$-5$

Q֐̒_̍W$i1C2j$łC_$i2C3j$ʂƂĈQ֐̕߂B

process
$y=a(x-1)^2+2$_$i2C3j$ʂ邩$3=a(2-1)^2+2$\hspace{8pt}$\therefore a=1$\\
$y=(x-1)^2+2=x^2-2x+3$

$y=(x-1)^2+2=x^2-2x+3$

Q֐$x$Ɠ_$i-1C0j$C$i1C0j$ŌC_$i0C1j$ʂƂĈQ֐̕߂B

process
$y=a(x+1)(x-1)$_$i0C1j$ʂ邩C
$1=a(0+1)(0-1)$\hspace{8pt}$\therefore a=-1$\\
$\therefore y=-(x+1)(x-1)=-x^2+1$

$y=-x^2+1$

$y=ax^2$̃Ot$(2,\,12)$ʂƂC̃Ot̎߂B

process
$12=a \times 2^2$C$\therefore a=3$

$y=3x^2$

_$(4,\,5)$𒸓_ƂC_$(6,\,-7)$ʂ2֐̎߂B

process
$y=a(x-4)^2+5$_$(6,\,-7)$ʂ邩C
$-7=a(6-4)^2+5$C$\therefore a=-3$B߂鎮
$y=-3(x-4)^2+5=-3x^2+24x-43$

$y=-3x^2+24x-43$

2֐$y=a(x+2)^2+1$i$a$͒萔j̃Ot_$(0,\,9)$ʂƂC
$a$̒l\fbox{\hspace{4pt} C \hspace{4pt}}łB

process
$9=a(0+2)^2+1$

$2$

2֐$y=ax^2-3x+1$i$a$͒萔j̃Ot_$(1,1)$ʂƂC
$a$̒l\fbox{\bf\hspace{4pt} C \hspace{4pt}}łB\\
\textcircled{\small 1}\,$1$~~~
\textcircled{\small 2}\,$2$~~~
\textcircled{\small 3}\,$3$~~~
\textcircled{\small 4}\,$4$

process
$1=a-3+1$~~~$\therefore a=3$

\textcircled{\small 3}

2֐$y=x^2-2x+k$i$k$͒萔j̃Ot_i3,\,2jʂƂC$k$̒l߂B

process
$2=3^2-2 \times 3+k$\\
$k=-1$

$k=-1$

2֐$y=a(x+1)^2-2ia͒萔j$̃Ot_i$0$C$1$jʂƂC
$a$̒l߂B

process
$1=a(0+1)^2-2$,\ $1=a-2$

$3$

2֐$y=a(x-1)^2+a$ia͒萔j̃Ot_$(0,\,4)$ʂƂC
$a$̒l߂B

process
$4=a(0-1)^2+a$C$a=2$

2

__ɂC_$(3,\,27)$ʂ2֐̎߂B

process
$y=ax^2$_$(3,\,27)$ʂ邩C$27=a\times 3^2$C$a=3$

$y=3x^2$

Ot̒_$(4,\,-3)$ŁC_$(3,\,-1)$ʂ2֐\fbox{\bf\,C\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$y=(x+4)^2-3$~~~
\textcircled{\small 2}\,$y=(x-4)^2-3$\\
\textcircled{\small 3}\,$y=2(x-4)^2-3$~~~
\textcircled{\small 4}\,$y=2(x+4)^2-3$

process
$y=a(x-4)^2-3$_$(3,\,-1)$ʂ邩C
$-1=a(3-4)^2-3$C$\therefore a=2$\\
$y=2(x-4)^2-3$

\textcircled{\small 3}

2֐$y=a(x-1)(x+3)$i$a$͒萔j̃Ot_$(0,\,-6)$ʂƂC
$a$̒l\fbox{\hspace{2pt} {\bf C} \hspace{2pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂IׁB\\
\textcircled{\small 1}\,$2$~~~
\textcircled{\small 2}\,$-2$~~~
\textcircled{\small 3}\,$3$~~~
\textcircled{\small 4}\,$-3$

process
$-6=a(0-1)(0+3)$

\textcircled{\small 1}

$y=3x^2$_$(-1,\,11)$C_$(1,\,-1)$ʂ悤ɕsړƂ2֐̎߂B

process
$y=3x^2+bx+c$_$(-1,\,11)$C_$(1,\,-1)$ʂ邩C\\
$11=3\times (-1)^2+b\times (-1)+c$C$\therefore -b+c=8\cdots$\textcircled{\small 1}\\
$-1=3\times 1^2+b\times 1+c$C$\therefore b+c=-4\cdots$\textcircled{\small 2}\\
\textcircled{\small 1}C\textcircled{\small 2}$b=-6,\,c=2$

$y=3x^2-6x+2$





[Level8]
ʂ_i3_bj
Q֐R_$(-1C2)$C$i0C1j$C$i1C-2j$ʂB

process
߂Q֐$y=ax^2+bx+c$ƂƁC\\
$\left\{ 
\begin{array}{l}
 2=a-b+c\cdots \textcircled{1}\\
 1=c\cdots \textcircled{2}\\
 -2=a+b+c\cdots \textcircled{3}
\end{array} \right.$ \\
\textcircled{1} $-$ \textcircled{2}F$1=a-b\cdots$\textcircled{4}\\
\textcircled{3} $-$ \textcircled{2}F$-3=a+b\cdots$\textcircled{5}\\
ƂɂC$a=-2\cdots$\textcircled{5}\\
\textcircled{4}C\textcircled{5}ɂC$a=-1Cb=-2$\\
āC߂Q֐
$y=-x^2-2x+1$

$y=-x^2-2x+1$

Q֐R_$(-1C-6)$C$i1C2j$C$i2C0j$ʂB

process
߂Q֐$y=ax^2+bx+c$ƂƁC\\
$\left\{ 
\begin{array}{l}
 -6=a-b+c\cdots \textcircled{1}\\
 2=a+b+c\cdots \textcircled{2}\\
 0=4a+2b+c\cdots \textcircled{3}
\end{array} \right.$ \\
\textcircled{2} $-$ \textcircled{1}F$8=2b\hspace{8pt} b=4\cdots$\textcircled{4}\\
\textcircled{3} $-$ \textcircled{2}F$-2=3a+b\hspace{8pt} $\textcircled{4}ƂɂC$a=-2\cdots$\textcircled{5}\\
\textcircled{2}C\textcircled{4}C\textcircled{5}ɂC$c=0$\\
āC߂Q֐
$y=-2x^2+4x$

$y=-2x^2+4x$














































[EOF]
% t@C̍Ōɂ
