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[Title]
212B_2jifunctionB

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% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
% tHg̑傫B1`10C܂TeX̃R}hw肷B
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% vAuɒǉpbP[Wt@Cw肷B
[usepackage]
\usepackage{color,amssymb}

% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
% 2sڈȍ~ɖƂ̉𓚂B
% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]
2֐̍őlEŏlibj
2֐$y=x^2-2x-3$
`悪C$-1\leqq x \leqq 4$łƂC
ŏlƂĐ̂C
\textcircled{\small 1}$\sim$
\textcircled{\small 6}̒1IсC
̔ԍ𓚂ȂB\\
\textcircled{\small 1}\,$y=-6\,(x=2̂Ƃ)$~~~
\textcircled{\small 2}\,$y=5\,(x=4̂Ƃ)$\\
\textcircled{\small 3}\,$y=0\,(x=-1̂Ƃ)$~~~
\textcircled{\small 4}\,$y=0\,(x=3̂Ƃ)$\\
\textcircled{\small 5}\,$y=-3\,(x=0̂Ƃ)$~~~
\textcircled{\small 6}\,$y=-4\,(x=1̂Ƃ)$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

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

\textcircled{\small 6}

2֐$y=x^2-3$ɂāC$x$̒`$-1 \leqq x \leqq 2$ƂƂC
$y$̍ől\fbox{\,A\,}$Cŏl\fbox{\,CE\,}$łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic50.tex}
\end{center}

process
$y$$x=2$̂ƂCől$y=2^2-3=1$ƂC
$y$$x=0$̂ƂCŏl$y=-3$ƂB

A:$1$CCE:$-3$

2֐$y=x^2+1$ɂāC$x$̒`$-1\leqq x \leqq 3$ƂƂC$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_$i0,\,1j$\\
őĺC$x=3$̂Ƃ$y=3^2+1=10$\\
ŏĺC$x=0$̂Ƃ$y=1$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic25.tex}
\end{center}

ől10i$x=3$̂Ƃj\\
ŏl1i$x=0$̂Ƃj

2֐$y=(x+1)^2+3$ɂāC$x$̒`$-3 \leqq x \leqq 2$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_$i-1,\,3j$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic26.tex}
\end{center}
őĺC$x=2$̂Ƃ$y=(2+1)^2+3=12$\\
ŏĺC$x=-1$̂Ƃ$y=3$

ől$12$i$x=2$̂ƂjC\\ ŏl$3$i$x=-1$̂Ƃj

2֐$y=(x-5)^2-9$ɂāC$x$̒`$0\leqq x \leqq 7$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_́i$5,\,-9$jBOtɂC\\
ől$x=0$̂Ƃ$y=(0-5)^2-9=16$C\\
ŏl$x=5$̂Ƃ$-9$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic27.tex}
\end{center}

ől$16$i$x=0$̂ƂjC\\ ŏl$-9$i$x=5$̂Ƃj

2֐$y=(x-3)^2+2$ɂāC$x$̒`$1 \leqq x \leqq 5$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic43.tex}
\end{center}
$x=1$̂ƂC$y=(1-3)^2+2=6$\\
$x=5$̂ƂC$y=(5-3)^2+2=6$\\
ől6i$x=1x=5̂Ƃ$jC\\
ŏl2i$x=3̂Ƃ$j

ől6i$x=1x=5̂Ƃ$jC\\
ŏl2i$x=3̂Ƃ$j

$y=\displaystyle\frac{1}{4} x^2$ɂāC$x$$-2\leqq x \leqq 4$͈̔͂̒lƂƂC
$y$̒l̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic63.tex}
\end{center}

ől$4$i$x=4$̂ƂjC\\
ŏl$0$i$x=0$̂Ƃj

$y=\displaystyle\frac{1}{4} x^2$ɂāC$x$$2\leqq x \leqq 6$͈̔͂̒lƂƂC
$y$̒l̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic130.tex}
\end{center}

ől$9$i$x=6$̂ƂjC\\
ŏl$1$i$x=2$̂Ƃj

2֐$y=x^2-2x-1$̒`悪$0\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-2)}{2\times 1}=1$C
_$y$W$=-2$BāC_$i1C-2j$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic117.tex}
\end{center}

ől2i$x=3$̂ƂjC\\ ŏl$-2$i$x=1$̂Ƃj

2֐$y=x^2-2x-2$ɂāC$x$̒`$-2 \leqq x \leqq 3$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=(x-1)^2-1^2-2=(x-1)^2-3$C
_$i1,\,-3j$\\
iʉj$=\displaystyle\frac{-(-2)}{2\times 1}=1$\\
_$y$W$=1^2-2\times 1-2=-3$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic75.tex}
\end{center}
őĺC$x=-2$̂Ƃ$y=(-2)^2-2(-2)-2=6$\\
ŏĺC$x=1$̂Ƃ$y=-3$

ől$6$i$x=-2$̂ƂjC\\ ŏl$-3$i$x=1$̂Ƃj

2֐$y=x^2-2x-2$ɂāC$x$̒`$2 \leqq x \leqq 4$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=(x-1)^2-1^2-2=(x-1)^2-3$C
_$i1,\,-3j$\\
iʉj$=\displaystyle\frac{-(-2)}{2\times 1}=1$\\
_$y$W$=1^2-2\times 1-2=-3$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic76.tex}
\end{center}
őĺC$x=4$̂Ƃ$y=4^2-2\times 4-2=6$\\
ŏĺC$x=2$̂Ƃ$y=-2$

ől$6$i$x=4$̂ƂjC\\ŏl$-2$i$x=2$̂Ƃj

֐$y=\displaystyle\frac{1}{2} x^2$ɂāC$x$̒`悪$-4\leqq x \leqq 2$̂Ƃ$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic79.tex}
\end{center}

ől8i$x=-4$̂ƂjC\\
ŏl0i$x=0$̂Ƃj  

֐$y=\displaystyle\frac{1}{2} x^2$ɂāC$x$̒`悪$2 \leqq x \leqq 6$̂Ƃ$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic131.tex}
\end{center}

ől18i$x=6$̂ƂjC\\
ŏl2i$x=2$̂Ƃj  

֐$y=x^2$ɂāC$x$̒`悪$-2\leqq x \leqq -1$̂Ƃ̕ω$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic80.tex}
\end{center}

ől4i$x=-2$̂ƂjC\\
ŏl1i$x=-1$̂Ƃj 

֐$y=x^2$ɂāC$x$̒`悪$-1\leqq x \leqq 3$̂Ƃ̕ω$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic129.tex}
\end{center}

ől9i$x=3$̂ƂjC\\
ŏl0i$x=0$̂Ƃj 

2֐$y=x^2-2x+2$̒`悪$-2\leqq x\leqq 0$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-2)}{2\times 1}=1$C
_$y$W$=1$BāC_i1C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic82.tex}
\end{center}

ől10i$x=-2$̂ƂjC\\ ŏl2i$x=0$̂Ƃj

2֐$y=x^2-2x+2$̒`悪$-1\leqq x\leqq 2$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-2)}{2\times 1}=1$C
_$y$W$=1$BāC_i1C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic83.tex}
\end{center}

ől5i$x=-1$̂ƂjC\\ ŏl1i$x=1$̂Ƃj

2֐$y=x^2-2x+2$̒`悪$0\leqq x\leqq 2$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-2)}{2\times 1}=1$C
_$y$W$=1$BāC_i1C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic84.tex}
\end{center}

ől2i$x=0C\,2$̂ƂjC\\ ŏl1i$x=1$̂Ƃj

$y=x^2-2x+2$i$0\leqq x\leqq 3$j̍őlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=x^2-2x$ƁC$y=x(x-2)$$x$02Ō邩C1B_$y$W$y=1^2-2\times 1+2=1$BāC_$i1C1j$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic101.tex}
\end{center}

ől5i$x=3$̂ƂjC\\ ŏl1i$x=1$̂Ƃj

2֐$y=x^2-2x+2$̒`悪$2\leqq x\leqq 4$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-2)}{2\times 1}=1$C
_$y$W$=1$BāC_i1C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic86.tex}
\end{center}

ől10i$x=4$̂ƂjC\\ ŏl2i$x=2$̂Ƃj

2֐$y=x^2+2$̒`悪$-3\leqq x\leqq -1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C2j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic87.tex}
\end{center}

ől11i$x=-3$̂ƂjC\\ ŏl3i$x=-1$̂Ƃj

2֐$y=x^2+2$̒`悪$-2\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C2j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic88.tex}
\end{center}

ől6i$x=-2$̂ƂjC\\ ŏl2i$x=0$̂Ƃj

2֐$y=x^2+2$̒`悪$-1\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C2j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic89.tex}
\end{center}

ől3i$x=-1,\,1$̂ƂjC\\ ŏl2i$x=0$̂Ƃj

2֐$y=x^2+2$̒`悪$1\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C2j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic90.tex}
\end{center}

ől11i$x=3$̂ƂjC\\ ŏl3i$x=1$̂Ƃj

2֐$y=x^2-1$̒`悪$-2\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C$-1$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic91.tex}
\end{center}

ől3i$x=-2$̂ƂjC\\ ŏl$-1$i$x=0$̂Ƃj

2֐$y=x^2-1$̒`悪$-2\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C$-1$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic127.tex}
\end{center}

ől8i$x=3$̂ƂjC\\ ŏl$-1$i$x=0$̂Ƃj

2֐$y=x^2-1$̒`悪$1\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C$-1$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic128.tex}
\end{center}

ől8i$x=3$̂ƂjC\\ ŏl$0$i$x=1$̂Ƃj

2֐$y=x^2-3$̒`悪$-2\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_i0C$-3$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic132.tex}
\end{center}

ől6i$x=3$̂ƂjC\\ ŏl$-3$i$x=0$̂Ƃj

2֐$y=x^2-4x+5$̒`悪$-1\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=1$BāC_i2C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic95.tex}
\end{center}

ől10i$x=-1$̂ƂjC\\ ŏl2i$x=1$̂Ƃj

2֐$y=x^2-4x+5$̒`悪$0\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=1$BāC_i2C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic96.tex}
\end{center}

ől5i$x=0$̂ƂjC\\ ŏl1i$x=2$̂Ƃj

2֐$y=x^2-4x+5$̒`悪$0\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=1$BāC_i2C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic147.tex}
\end{center}

ől5i$x=0$̂ƂjC\\ ŏl2i$x=1$̂Ƃj

2֐$y=x^2-4x+5$̒`悪$1\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=1$BāC_i2C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic97.tex}
\end{center}

ől2i$x=1,\,3$̂ƂjC\\ ŏl1i$x=2$̂Ƃj

2֐$y=x^2-4x+5$̒`悪$3\leqq x\leqq 5$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=1$BāC_i2C1j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic98.tex}
\end{center}

ől10i$x=5$̂ƂjC\\ ŏl2i$x=3$̂Ƃj

2֐$y=x^2+2x$̒`悪$-2\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=x(x+2)$ƈł邩C2֐$x$$x=0x=-2$ŌBāC
$=\displaystyle\frac{-2}{2\times 1}=-1$C
_$y$W$=-1$BāC_i$-1$C$-1$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic99.tex}
\end{center}

ől3i$x=1$̂ƂjC\\ ŏl$-1$i$x=-1$̂Ƃj

2֐$y=x^2-6x+3$̒`悪$0\leqq x\leqq 6$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-6)}{2\times 1}=3$C
_$y$W$=-6$BāC_i$3$C$-6$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic100.tex}
\end{center}

ől3i$x=0,\,6$̂ƂjC\\ ŏl$-6$i$x=3$̂Ƃj

2֐$y=x^2-4x+7$̒`悪$-1\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
āC_$\left(2C\,3 \right)$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic107.tex}
\end{center}

ől12i$x=-1$̂ƂjC\\ 
ŏl4i$x=1$̂Ƃj

2֐$y=x^2-4x+7$̒`悪$0\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
āC_$\left(2C\,3 \right)$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic141.tex}
\end{center}

ől7i$x=0$̂ƂjC\\ 
ŏl3i$x=2$̂Ƃj

2֐$y=x^2-6x+5$ɂāC$x$̒`$1 \leqq x \leqq 4$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=x^2-6x$$y=x(x-6)$ƈłC$x$$x=0$$x=6$ŌB
āC$y=x^2-6x+5$̒_$(3,\ -4)$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic133.tex}
\end{center}
őĺC$x=1$̂Ƃ$y=1^2-6\times 1+5=0$\\
ŏĺC$x=3$̂Ƃ$y=-4$

ől$0$i$x=1$̂ƂjC\\ŏl$-4$i$x=3$̂Ƃj

2֐$y=x^2-6x+5$ɂāC$x$̒`$0 \leqq x \leqq 2$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=x^2-6x$$y=x(x-6)$ƈłC$x$$x=0$$x=6$ŌB
āC$y=x^2-6x+5$̒_$(3,\ -4)$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic145.tex}
\end{center}
őĺC$x=0$̂Ƃ$y=0^2-6\times 0+5=5$\\
ŏĺC$x=2$̂Ƃ$y=-3$

ől$5$i$x=0$̂ƂjC\\ŏl$-3$i$x=2$̂Ƃj

2֐$y=x^2-2x-3$ɂāC$x$̒`$-2 \leqq x \leqq 2$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=x^2-2x$$y=x(x-2)$ƈłC$x$$x=0$$x=2$ŌB
āC$y=x^2-2x-3$̒_$(1,\ -4)$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic134.tex}
\end{center}
őĺC\\ $x=-2$̂Ƃ$y=(-2)^2-2\times (-2)-3=5$\\
ŏĺC$x=1$̂Ƃ$y=-4$

ől$5$i$x=-2$̂ƂjC\\ŏl$-4$i$x=1$̂Ƃj

2֐$y=x^2+5$̒`悪$-2\leqq x\leqq 1$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_$(0,\,5)$B
$x=-2$̂Ƃől$y=(-2)^2+5=9$\\
$x=0$̂Ƃŏl$5$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2701m0401.tex}
\end{center}

ől9i$x=-2$̂ƂjC\\ ŏl5i$x=0$̂Ƃj

2֐
$y=(x-3)^2+5$
`悪$2\leqq x\leqq 5$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_$(3,\,5)$B\\
$x=5$̂Ƃől$y=(5-3)^2+5=9$\\
$x=3$̂Ƃŏl$5$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2601m0401.tex}
\end{center}

ől9i$x=5$̂ƂjC\\ ŏl5i$x=3$̂Ƃj

2֐$y=x^2+2x-3$̒`悪$0\leqq x\leqq 2$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-2}{2\times 1}=-1$C
_$y$W$=-5$BāC_$i-1C-5j$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic142.tex}
\end{center}

ől5i$x=2$̂ƂjC\\ ŏl$-3$i$x=0$̂Ƃj

2֐$y=x^2-4x+3$̒`悪$0\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$=\displaystyle\frac{-(-4)}{2\times 1}=2$C
_$y$W$=-1$BāC_$i2C-1j$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic143.tex}
\end{center}

ől3i$x=0$̂ƂjC\\ ŏl$-1$i$x=2$̂Ƃj

2֐$y=x^2-2x-5$ɂāC$x$̒`$0 \leqq x \leqq 4$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$y=(x-1)^2-1^2-5=(x-1)^2-6$C
_$i1,\,-6j$\\
iʉj$=\displaystyle\frac{-(-2)}{2\times 1}=1$\\
_$y$W$=1^2-2\times 1-5=-6$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic146.tex}
\end{center}
őĺC$x=4$̂Ƃ$y=4^2-2\times 4-5=3$\\
ŏĺC$x=1$̂Ƃ$y=-6$

ől$3$i$x=4$̂ƂjC\\ŏl$-6$i$x=1$̂Ƃj

2֐$y=(x-2)^2-5$ɂāC$x$̒`$-3 \leqq x \leqq 3$ƂƂC
$y$̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
_$i2,\,-5j$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m0303.tex}
\end{center}
őĺC$x=-3$̂Ƃ$y=(-3-2)^2-5=20$\\
ŏĺC$x=2$̂Ƃ$y=-5$

ől$20$i$x=-3$̂ƂjC\\ŏl$-5$i$x=2$̂Ƃj

$y=-3(x-2)^2+1$ɂāC
$x$̕ψ$0\leqq x \leqq 3$ƂƂC
$y$̒l̍őlƍŏlƂ̂Ƃ$x$̒l߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxminminus.tex}
\end{center}

ől$1$i$x=2$̂ƂjC\\
ŏl$-11$i$x=0$̂Ƃj

2֐$y=x^2-2mx+3m$̍ŏl$k$ƂB
$k$ł傫ȂƂ$m$̒l\fbox{\hspace{8pt} 9 \hspace{8pt}}łB\\
\textcircled{\small 1}\,$1$~~~
\textcircled{\small 2}\,$\displaystyle\frac{3}{2}$~~~
\textcircled{\small 3}\,$\displaystyle\frac{9}{4}$~~~
\textcircled{\small 4}\,$3$

process
$y=x(x-2m)+3m$̒_$(m,\,3m-m^2)$ł邩C$k=3m-m^2=m(3-m)$$m=\displaystyle\frac{3}{2}$̂ƂőƂȂB

\textcircled{\small 2}

2֐$y=-x^2+k$i$k$͒萔jɂāC
$x$̕ψ$-1\leqq x \leqq 2$ƂƂC
$y$̍ŏl$-1$łB
̂ƂC$k$̒l\fbox{\,\bf A\,}łB

process
$-1=f(2)=-2^2+k$C$\therefore k=3$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2902401.tex}
\end{center}

$3$

2֐$y=(x+1)^2+5$ɂāC$x$̕ψ$-3\leqq x\leqq 1$ƂƂC
$y$̍ől\fbox{\,\bf A\,}Cŏl\fbox{\,\bf C\,}łB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801401.tex}
\end{center}

AF$9$CCF$5$

2֐$y=-(x+3)^2+2$ɂāC$x$̕ψ$-4 \leqq x \leqq 0$ƂƂC
$y$̍őlэŏlƂ̂Ƃ$x$̒l߂B
\begin{center}\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxminminus.tex}\end{center}

ől$2$i$x=-3$̂ƂjC\\
ŏl$-7$i$x=0$̂Ƃj

2֐
$y=-(x-1)^2+2$̍őlƍŏlɂĂ̋LqƂāC
̂
\fbox{\bf\hspace{4pt} A \hspace{4pt}}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂IׁB\\
\textcircled{\small 1}\,$x=1$ōől$2$ƂCŏl͂ȂB\\
\textcircled{\small 2}\,$x=1$ōŏl$2$ƂCől͂ȂB\\
\textcircled{\small 3}\,$x=-1$ōől$2$ƂCŏl͂ȂB\\
\textcircled{\small 4}\,$x=-1$ōŏl$2$ƂCől͂ȂB

process
_$(1,\,2)$Cɓ

\textcircled{\small 1}

2֐$y=(x-2)^2+5$ɂāC
$x$̕ψ$0 \leqq x \leqq 3$ƂƂ
$y$̍ől\fbox{\bf\, A \,}Cŏl\fbox{\bf\, C \,}łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic_maxmin.tex}
\end{center}

process
$x=0$̂Ƃől$9$C
$x=2$̂Ƃŏl$5$

AF$9$CCF$5$







[Level2]
2֐̍őlEŏlibj
2֐$y=x^2-3x+4$̒`悪$0\leqq x\leqq 4$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B

process
$=\displaystyle\frac{-(-3)}{2\times 1}=\displaystyle\frac{3}{2}$C
āC_$\left(\displaystyle\frac{3}{2}C\,\displaystyle\frac{7}{4} \right)$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic106.tex}
\end{center}

ől8i$x=4$̂ƂjC\\ ŏl$\displaystyle\frac{7}{4}\,\left(x=\displaystyle\frac{3}{2}̂Ƃ\right)$

2֐$y=2(x+1)^2+k$ik͒萔jɂāC$x$̕ϐ$-4\leqq x \leqq 0 $ƂƂC
$y$̍ől15łB̂ƂC$k$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic54.tex}
\end{center}
$x=-1$牓Ă̂$x=-4$B
āC$x=-4$̂Ƃől$15$ł邩C\\
$15=2(-4+1)^2+k$C$k=-3$

$k=-3$

2֐$y=(x-3)^2+kik͒萔j$ɂāC
$x$̕ψ$0\leqq x \leqq 4$ƂƂC
$y$̍ől$14$łB
̂Ƃ$k$̒l߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic58.tex}
\end{center}
$x=3$牓Ă̂$x=0$B
$x=0$̂Ƃ$y=14$ł邩C\\
$14=(0-3)^2+k$C$k=5$

5

2֐$y=-x^2+4$ɂāC$x$̕ψ$-2 \le x \le 1$ƂƂC
$y$̕ψ$\fbox{\,A\,}\le y \le \fbox{\,C\,}$łB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic37.tex}
\end{center}

$0 \le y \le 4$

2֐$y=-(x-3)^2+4$ɂāC$x$̕ψ$2\leqq x\leqq 5$ƂƂC
$y$̍őlƍŏl߂B

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic60.tex}
\end{center}
$x=3$牓Ă̂$x=5$B\\
āC $x=5$̂Ƃŏl$y=-(5-3)^2+4=0$

ől$4ix=3̂Ƃj$C\\
ŏl$0ix=5̂Ƃj$

2֐$y=-2(x+1)^2+k$i$k$͒萔jɂāC
$x$̕ψ$-2 \le x \le 1$ƂƂC$y$̍ŏl$4$łB
̂ƂC$k$̒l߂B

process
$x=-1$̕$1$ւ̕傫C$y$$x=1$̂Ƃŏl$4$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic34.tex}
\end{center}
$4=-2(1+1)^2+k$, \ $4=-2 \times 4 +k$

$k=12$

2֐$y=-(x-2)^2-1$ɂāC$x$̕ψ$1\le x \le 4$ƂƂC
$y$̕ψ\fbox{AC} $\le$ \fbox{EG}łB

process
}C$-5 \le y \le -1$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic46.tex}
\end{center}

$-5 \le y \le -1$

֐$y=-2x^2$ɂāC$x$̕ψ悪$-1\leqq x \leqq 2$̂Ƃ̕ω$y$̕ψ߂B\\

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic81.tex}
\end{center}

$-8 \leqq y \leqq 0$

2֐$y=2x^2-3$̒`悪$2\leqq x\leqq 4$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B

process
_i0C$-3$j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic92.tex}
\end{center}

ől29i$x=4$̂ƂjC\\ ŏl$5$i$x=2$̂Ƃj

2֐$y=-x^2+2$̒`悪$-1\leqq x\leqq 2$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B

process
_i0C2j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic93.tex}
\end{center}

ől2i$x=0$̂ƂjC\\ ŏl$-2$i$x=2$̂Ƃj

2֐$y=-2x^2+8$̒`悪$1\leqq x\leqq 3$̂ƂCőlэŏlƂ̂Ƃ$x$̒l߂B

process
_i0C8j\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic94.tex}
\end{center}

ől6i$x=1$̂ƂjC\\ ŏl$-10$i$x=3$̂Ƃj

2֐$y=-x^2-6x-3$̍őlƍŏl߂B
C$-5 \leqq x \leqq 1$ƂB

process
$y=-x^2-6x-3=-(x^2+6x)-3=-(x+3)^2+9-3=-(x+3)^2+6$\\
iʉj$x=\displaystyle\frac{-(-6)}{2\times (-1)}=-3$\\
_$y$W$=(-3)^2-6\times (-3)-3=6$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic65.tex}
\end{center}

ől$6$i$x=-3$̂ƂjC\\
ŏl$-10$i$x=1$̂Ƃj

$y=2x^2+4x+1$̍ŏl߂B

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

ŏl$-1$i$x=-1$̂Ƃj

$y=-2x^2-4x+1$i$-2\leqq  x<1$j̍őlCŏl߂B

process
$y=-2(x^2+2x)+1=-2(x+1)^2+2\times 1^2+1\\
=-2(x+1)^2+3$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic102.tex}
\end{center}

őlF3i$x=-1$̂ƂjCŏlFȂ

$y=x^2+4x+1$i$-4<x\leqq -1$j̍őlэŏl߂B

process
$y=x^2+4x$ƁC$y=x(x+4)$ƂȂ邩C$x$0$-4$ŌBāC$x=-2$B_$y$W$y=(-2)^2+4\times (-2) +1=-3$B_$i-2C-3j$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic103.tex}
\end{center}

őlFȂCŏlF$-3$i$x=-2$̂Ƃj

2֐$y=-(x-3)^2+k$i$k$͒萔jɂāC$x$̒`
$-2\leqq x\leqq 4$ƂƂC
$y$̍ŏl$-20$łB
̂ƂC$k$̒l$\fbox{\bf\,A\,}$łB

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2702m0401.tex}
\end{center}
$x=-2$̂ƂŏlƂB\\
$-20=-(-2-3)^2+k$~~~$-20=-25+k$

$5$

2֐
$y=-(x-1)^2+k$i$k$͒萔jɂāC
$x$̕ψ$0\leqq x\leqq 4$ƂƂC
$x$̍ŏl$-5$łB
̂ƂC$k$̒l\fbox{\bf \,A\,}łB

process
$x=4$̂ƂɍŏƂȂB\\
$-5=-(4-1)^2+k~~~\therefore k=4$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2602m0401.tex}
\end{center}

4





[Level3]
őEŏ͑̕ibj
20cm̐j܂ȂĒ`B`̏c$x$cmƂāCʐ$y\,{\rm cm^2}$̍ől߂B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic77.tex}
\end{center}

process
`̉̒$(10-x)$cmƕ\BCӂ̒͐ł邩C$0<x<10$łB̂ƂC`̖ʐ$y\,{\rm cm^2}$\\
$y=x(10-x)$ƂȂB̃Ot$x$010Ō邩C$x=5$łCOť`͏ɓʂł邩C$y$$x=5$̂Ƃől25ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic78.tex}
\end{center}

ől25${\rm cm^2}$

ӂ̒ƍ̘a6łOp`̖ʐς̍ől߂B

process
ӂ̒$x$ƂƁC$6-x$ł邩COp`̖ʐς$y$Ƃ\\
$y=\displaystyle\frac{1}{2} x(6-x)$\\
܂C$x>0$C$6-x>0$ł邩C$0<x<6$\\
āCʐς$x=3$̂ƂCől\\
$y=\displaystyle\frac{1}{2}\times 3\times (6-3)=\displaystyle\frac{9}{2}$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic104.tex}
\end{center}

$\displaystyle\frac{9}{2}$

̒8ł钷`̖ʐς̍ől߂B

process
Pӂ̒$x$ƂƁCאڂӂ̒$4-x$ƂB
āC`̖ʐ$y$́C\\
$y=x(4-x)$ƕ\ƂłB̊֐$x$04Ō邩C$x=2$łB
$y$$x=2$̂Ƃől$y=2(4-2)=4$ƂB

őlF4iӂ̒2̐`̂Ƃj

p͂2ӂ̘̒a${\rm 20cm}$ł悤ȒpOp`B
̒pOp`̖ʐ$y {\rm cm}^2$̍ől߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic148.tex}
\end{center}

process
$y=\displaystyle\frac{1}{2} x(20-x)$$x=10$̂ƂɍőƂȂ邩C
ől$y=\displaystyle\frac{1}{2} \times 10\times 10=50$

$50$

p͂2ӂ̘̒a${\rm 8cm}$ł悤ȒpOp`B
̒pOp`̖ʐ$y {\rm cm}^2$̍ől߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic148b.tex}
\end{center}

process
$y=\displaystyle\frac{1}{2} x(8-x)$$x=4$̂ƂɍőƂȂ邩C
ől$y=\displaystyle\frac{1}{2} \times 4\times 4=8$

$8$

p͂2ӂ̘̒a${\rm 16cm}$ł悤ȒpOp`B
̒pOp`̖ʐ$y {\rm cm}^2$̍ől߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic148c.tex}
\end{center}

process
$y=\displaystyle\frac{1}{2} x(16-x)$$x=8$̂ƂɍőƂȂ邩C
ől$y=\displaystyle\frac{1}{2} \times 8\times 8=32$

$32$





[Level4]
2֐̃Otx̋L_̍Wibj
2֐$y=x^2+x-6$̃Ot$x$Ƃ̋L_$x$ẂC\\
i\fbox{ GI },\,0j,i\fbox{ J },\,0j\\
łB

process
2֐$y=x^2+x-6$$x$i$y=0$j̋L_$x$ẂC\\
$0=x^2+x-6=(x+3)(x-2)$\\
$x=-3,\,2$

$-3,2$

2֐$y=2x^2+x-15$̃Ot$x$Ƃ̋L_̍ẂC\\
$\left(\fbox{ IJ },\,0 \right)$,$\left(\displaystyle \frac{\fbox{ L }}{\fbox{ N }},\,0 \right)$\\
łB

process
2֐$y=2x^2+x-15$$x$i$y=0$j̋L_$x$ẂC\\
$0=2x^2+x-15=(x+3)(2x-5)$\\
$x=-3,\,\displaystyle \frac{5}{2}$

$-3,\displaystyle \frac{5}{2}$

2֐$y=2x^2-7x+4$̃Ot$x$Ƃ̋L_$x$W߂B

process
$x$i$y=0$jƂ̌_$x$ẂC\\
$0=2x^2-7x+4$̉B\\
$a=2,\,b=-7,\,c=4$\\
$x=\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}\\
=\displaystyle \frac{-(-7)\pm \sqrt{(-7)^2-4 \times 2 \times 4}}{2 \times 2}\\
=\displaystyle \frac{7\pm \sqrt{49-32}}{4}=\displaystyle \frac{7\pm \sqrt{17}}{4}$

$\displaystyle \frac{7\pm \sqrt{17}}{4}$

2֐$y=x^2-5x+2$̃Ot$x$Ƃ̋L_$x$W߂B

process
$x$i$y=0$jƂ̋L_$x$ẂC\\
$0=x^2-5x+2$ŁC$a=1Cb=-5Cc=2$ƂāC\\
$x=\displaystyle \frac{-(-5)\pm \sqrt{(-5)^2-4 \times 1 \times 2}}{2 \times 1}
=\displaystyle \frac{5 \pm \sqrt{17}}{2}$

$\displaystyle \frac{5 \pm \sqrt{17}}{2}$

2֐$y=3x^2+14x-5$̃Ot$x$Ƃ̋L_$x$ẂC
$\left( \fbox{EG},\,0 \right),\,\left( \displaystyle\frac{\fbox{I}}{\fbox{J}},\,0 \right)$łB

process
$y=3x^2+14x-5=0$ƁC
$b'=7$C$x=\displaystyle\frac{-7\pm \sqrt{7^2-3\times (-5)}}{3}=\displaystyle\frac{-7\pm 8}{3}
=-5,\,\displaystyle\frac{1}{3}$

$-5,\displaystyle\frac{1}{3}$

2֐$y=3x^2+2x-8$̃Ot$x$Ƃ̋L_$x$W
$\left( \fbox{CE},\,0 \right),\,\left( \displaystyle\frac{\fbox{G}}{\fbox{I}},\,0 \right)$łB

process
$b'=1,\,x=\displaystyle\frac{-1\pm \sqrt{1^2-3\times (-8)}}{3}=\displaystyle\frac{-1\pm 5}{3}
=-2,\,\displaystyle\frac{4}{3}$

$-2,\displaystyle\frac{4}{3}$

$y=x^2-2x-3$ɂāC$x$Ƃ̋L_$x$W߂ȂB

process
$0=x^2-2x-3$ƁC$0=(x+1)(x-3)$ƈł邩C
$x=-1,\,3$

$x=-1,\,3$

2֐
$y=5x^2-9x-2$
̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=5x^2-9x-2$ƁC\\
$0=(5x+1)(x-2)~~~\therefore x=-\displaystyle\frac{1}{5},\,2$

$-\displaystyle\frac{1}{5},\,2$

2֐
$y=3x^2+4x-4$
̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=3x^2+4x-4$ƁC\\
$0=(3x-2)(x+2)~~~x=\displaystyle\frac{2}{3},\,-2$

$\displaystyle\frac{2}{3},\,-2$

2֐$y=x^2-3x+1$̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=x^2-3x+1$\\
$x=\displaystyle\frac{3\pm\sqrt{(-3)^2-4\cdot 1 \cdot 1}}{2\cdot 1}
=\displaystyle\frac{3\pm \sqrt{5}}{2}$

$x=\displaystyle\frac{3\pm \sqrt{5}}{2}$

2֐
$y=5x^2-7x+2$
̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=5x^2-7x+2$ƁC\\
$0=(5x-2)(x-1)~~~x=1,\,\displaystyle\frac{2}{5}$

$1,\,\displaystyle\frac{2}{5}$

2֐$y=x^2-5x+5$̃Ot$x$Ƃ̋L_$x$W߂B

process
$x=\displaystyle\frac{5\pm \sqrt{25-20}}{2}=\displaystyle\frac{5\pm \sqrt{5}}{2}$

$\displaystyle\frac{5\pm \sqrt{5}}{2}$

2֐$y=3x^2-10x+3$̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=3x^2-10x+3=(3x-1)(x-3)$

$x=3,\displaystyle\frac{1}{3}$

2֐$y=2x^2+3x-5$̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=2x^2+3x-5=(2x+5)(x-1)$

$x=1,-\displaystyle\frac{5}{2}$

2֐$y=x^2-4x-5$̃Ot$x$Ƃ̋L_$x$W߂B

process
$0=x^2-4x-5=(x+1)(x-5)$

$x=-1,5$

2֐$y=2x^2+x-1$̃Ot$x$Ƃ̋L_̍ẂC\\
$\left(\fbox{ GI },\,0 \right)$,$\left(\displaystyle \frac{\fbox{ J }}{\fbox{ L }},\,0 \right)$\\
łB

process
2֐$y=2x^2+x-1$$x$i$y=0$j̋L_$x$ẂC\\
$0=2x^2+x-1=(x+1)(2x-1)$\\
$x=-1,\,\displaystyle \frac{1}{2}$

$-1,\displaystyle \frac{1}{2}$

2֐$y=3x^2-4x-7$̃Ot
$x$Ƃ̋L_̍ẂC
$\left( \fbox{\bf\,EG,0 \,} \right),\left( \displaystyle\frac{\fbox{\bf\, I\,}}{\fbox{\bf\, J\,}},0 \right)$
łB

process
$0=3x^2-4x-7=(3x-7)(x+1)$\\
$\therefore x=-1,\displaystyle\frac{7}{3}$

EGF$-1$CIJF$\displaystyle\frac{7}{3}$

2֐$y=x^2+3x+1$̃Ot$x$Ƃ̋L_$x$ẂC
$x=\displaystyle\frac{\fbox{\bf\, CE \,}\pm\sqrt{\fbox{\bf\, G \,}}}{\fbox{\bf\, I \,}}$
łB

process
$x=\displaystyle\frac{-3\pm\sqrt{9-4}}{2}=\displaystyle\frac{-3\pm\sqrt{5}}{2}$

$\displaystyle\frac{-3\pm\sqrt{5}}{2}$

2֐$y=x^2-2x-3$
Ot$x$Ƃ̌_̍WƂ
̂C
\textcircled{\small 1}$\sim$
\textcircled{\small 6}̒1IсC
̔ԍ𓚂ȂB\\
\textcircled{\small 1}\,$(-3,0),\,(1,0)$~~~
\textcircled{\small 2}\,$(-3,0),\,(-1,0)$~~~
\textcircled{\small 3}\,$(-1,0),\,(3,0)$\\
\textcircled{\small 4}\,$(1,0),\,(3,0)$~~~
\textcircled{\small 5}\,$(-3,0),\,(2,0)$~~~
\textcircled{\small 6}\,$(-1,0),\,(2,0)$

process
$0=(x+1)(x-3)$C$x=-1,3$

\textcircled{\small 3}








[Level5]
2֐̃Otx̋L_̐ibj
2֐$y=x^2+8x+k$i$k$͒萔j̃Ot$x$ɐڂƂC$k$̒l߂B

process
$0=x^2+8x+k$
̔ʎ$D$ƂƁC$D=8^2-4 \times 1 \times k=0$ł΂悢B\\
$k=16$\\
iʁj\,
$y=(x+4)^2-4^2+k=(x+4)^2-16+k$̒_$(-4,k-16)$C$k-16=0$$k=16$\\

$k=16$

2֐$y=x^2+6x+k$i$k$͒萔j̃Ot$x$ƐڂƂC
$k$̒l߂B

process
$0=y=x^2+6x+k$̔ʎ$D$ƂƁC
C\\
$D=6^2-4 \times 1 \times k=0$
@$k=9$\\
iʉj\\
$xiy=0j$ƐڂƂC2֐\\
$y=(x+3)^2$ƊSɕό`͂ł邩C
̂Q֐WJ\\
$y=x^2+6x+9$

$9$

2֐$y=x^2+6x+9$̃Ot$x$Ƃ̋L_̌͉B

process
$0=x^2+6x+9$
ʎ$D$ƂƁC\\
$D/4=3^2-1\times 9=0$1B\\
iʁj\,
$y=(x+3)^2-3^2+9=(x+3)^2$B\\
Ot͐}̂悤ɂȂC
2֐$x$͐ڂĂ邩狤L_1B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic35.tex}
\end{center}

$1$

2֐$y=2x^2+8x+mim͒萔j$̃Ot$x$ɐڂƂC$m$̒l߂B

process
$0=2x^2+8x+m$
ʎ$D$ƂƁC\\
$D/4=4^2-2m=0$~~~$m=8$\\
iʁj\,
$2x^2+8x+m=0i\Longleftrightarrow m=-2x^2-8xj$̉̌$1$B
āC$y=my=-2x^2-8x$̋L_̌1ł΂悢B\\
$y=-2x^2-8x=-2(x^2+4x)=-2(x+2)^2+8$ƕό`ł邩COt
$m=8$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic61.tex}
\end{center}

8

2֐
$y=x^2-2x-1$
̃Ot$x$Ƃ̋L_̌\fbox{\bf\,C\,}łB

process
$0=x^2-2x-1$̔ʎ$D$ƂƁC
$\displaystyle\frac{D}{4}=(-1)^2-1\times (-1)=2>0$\\
iʁj$D=(-2)^2-4\times 1\times (-1)=8>0$

2

2$x^2-6x+a=0$
1̎ƂC萔$a$̒l$a=$
\fbox{\hspace{8pt}13\hspace{8pt}}łB

process
ʎ$D$ƂƁC\\
$0=D=(-6)^2-4 \cdot 1\cdot a$~~~$\therefore a=9$\\
iʁj\,$0=D/4=(-3)^2-1 \cdot a$~~~$\therefore a=9$

$9$

2$x^2-4x+a=0$
1̎ƂC萔$a$̒l$a=$
\fbox{\hspace{8pt}13\hspace{8pt}}łB

process
ʎ$D$ƂƁC\\
$0=D=(-4)^2-4 \cdot 1\cdot a$~~~$\therefore a=4$\\
iʁj\,$0=D/4=(-2)^2-1 \cdot a$~~~$\therefore a=4$

$4$

2$x^2-2x+a=0$
1̎ƂC萔$a$̒l$a=$
\fbox{\hspace{8pt}13\hspace{8pt}}łB

process
ʎ$D$ƂƁC\\
$0=D=(-2)^2-4 \cdot 1\cdot a$~~~$\therefore a=1$\\
iʁj\,$0=D/4=(-1)^2-1 \cdot a$~~~$\therefore a=1$

$1$

2$2x^2-4x+a=0$
1̎ƂC萔$a$̒l$a=$
\fbox{\hspace{8pt}13\hspace{8pt}}łB

process
ʎ$D$ƂƁC\\
$0=D/4=(-2)^2-2\times a$

$2$






[Level6]
2sibIj
2s$x^2+4x-5 \leqq 0$B\\
\textcircled{\small 1}\,$-5 \leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x \leqq -5, 1 \leqq x$\\
\textcircled{\small 3}\,$-1 \leqq x \leqq 5$~~~
\textcircled{\small 4}\,$x \leqq  -1,5  \leqq x$

process
$(x+5)(x-1) \leqq 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic31.tex}
\end{center}

\textcircled{\small 1}

2s$x^2+4x-5 \geqq 0$B\\
\textcircled{\small 1}\,$-5 \leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x \leqq -5, 1 \leqq x$\\
\textcircled{\small 3}\,$-1 \leqq x \leqq 5$~~~
\textcircled{\small 4}\,$x \leqq  -1,5  \leqq x$

process
$(x+5)(x-1) \geqq 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic31.tex}
\end{center}

\textcircled{\small 2}

2s$x^2-10x+21\leqq 0$B

process
$(x-3)(x-7) \leqq 0$ \\
\textcircled{\small 1}\,$-7 \leqq x \leqq -3$~~~
\textcircled{\small 2}\,$x \leqq -7, -3 \leqq x$\\
\textcircled{\small 3}\,$3 \leqq x \leqq 7$~~~
\textcircled{\small 4}\,$x \leqq  3,7  \leqq x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic32.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-10x+21\geqq 0$B\\
\textcircled{\small 1}\,$-7 \leqq x \leqq -3$~~~
\textcircled{\small 2}\,$x \leqq -7, -3 \leqq x$\\
\textcircled{\small 3}\,$3 \leqq x \leqq 7$~~~
\textcircled{\small 4}\,$x \leqq  3,7  \leqq x$

process
$(x-3)(x-7) \geqq 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic32.tex}
\end{center}

\textcircled{\small 4}

2s$x^2+4x \geqq 0$B\\
\textcircled{\small 1}\,$-4 \leqq x \leqq 0$~~~
\textcircled{\small 2}\,$x \leqq -4, 0 \leqq x$\\
\textcircled{\small 3}\,$0 \leqq x \leqq 4$~~~
\textcircled{\small 4}\,$x \leqq  0,4  \leqq x$

process
$x(x+4) \geqq 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic33.tex}
\end{center}

\textcircled{\small 2}

2s$x^2+4x \leqq 0$B\\
\textcircled{\small 1}\,$-4 \leqq x \leqq 0$~~~
\textcircled{\small 2}\,$x \leqq -4, 0 \leqq x$\\
\textcircled{\small 3}\,$0 \leqq x \leqq 4$~~~
\textcircled{\small 4}\,$x \leqq  0,4  \leqq x$

process
$x(x+4) \leqq 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic33.tex}
\end{center}

\textcircled{\small 1}

2s$x^2-x-2\leqq 0$B\\
\textcircled{\small 1}\,$-2 \leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x \leqq -2, 1 \leqq x$\\
\textcircled{\small 3}\,$-1 \leqq x \leqq 2$~~~
\textcircled{\small 4}\,$x \leqq  -1,2  \leqq x$

process
$(x+1)(x-2)\leqq 0$ \\
2֐$y=(x+1)(x-2)$$x$Ƃ̌_$x$ẂC$-1$$2$łB\\
}ɂC$-1 \leqq x \leqq 2$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic36.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-x-2\geqq 0$B\\
\textcircled{\small 1}\,$-2 \leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x \leqq -2, 1 \leqq x$\\
\textcircled{\small 3}\,$-1 \leqq x \leqq 2$~~~
\textcircled{\small 4}\,$x \leqq  -1,2  \leqq x$

process
$(x+1)(x-2)\geqq 0$ \\
2֐$y=(x+1)(x-2)$$x$Ƃ̌_$x$ẂC$-1$$2$łB\\
}ɂC$x \leqq -1C2 \leqq x $
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic36.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-4x-5<0$ȂB
C}́C2֐$y=x^2-4x-5$̃OtłB\\
\textcircled{\small 1}\,$-5 < x < 1$~~~
\textcircled{\small 2}\,$x < -5, 1 < x$\\
\textcircled{\small 3}\,$-1 < x < 5$~~~
\textcircled{\small 4}\,$x <  -1,5  < x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic39.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic39p.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-4x-5>0$ȂB
C}́C2֐$y=x^2-4x-5$̃OtłB\\
\textcircled{\small 1}\,$-5 < x < 1$~~~
\textcircled{\small 2}\,$x < -5, 1 < x$\\
\textcircled{\small 3}\,$-1 < x < 5$~~~
\textcircled{\small 4}\,$x <  -1,5  < x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic39.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic39p.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-2x<0$B
}́C2֐$y=x^2-2x$̃OtłB\\
\textcircled{\small 1}\,$-2 < x < 0$~~~
\textcircled{\small 2}\,$x < -2, 0 < x$\\
\textcircled{\small 3}\,$0 < x < 2$~~~
\textcircled{\small 4}\,$x <  0,2  < x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic44.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-2x>0$B
}́C2֐$y=x^2-2x$̃OtłB\\
\textcircled{\small 1}\,$-2 < x < 0$~~~
\textcircled{\small 2}\,$x < -2, 0 < x$\\
\textcircled{\small 3}\,$0 < x < 2$~~~
\textcircled{\small 4}\,$x <  0,2  < x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic44.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-x-6>0$B
C}́C2֐$y=x^2-x-6$̃OtłB\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$\\
\textcircled{\small 5}\,$-3<x<-2$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic51.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-x-6 < 0$B\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$\\
\textcircled{\small 5}\,$-3<x<-2$

process
$(x+2)(x-3)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic109.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-4<0$B\\
\textcircled{\small 1}\,$-2<x<2$~~~
\textcircled{\small 2}\,$x<-2,2<x$\\
\textcircled{\small 3}\,$x\not= -2 $~~~
\textcircled{\small 4}\,$x\not= 2$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic55.tex}
\end{center}

\textcircled{\small 1}

2s$x^2-4>0$B\\
\textcircled{\small 1}\,$-2<x<2$~~~
\textcircled{\small 2}\,$x<-2,2<x$\\
\textcircled{\small 3}\,$x\not= -2 $~~~
\textcircled{\small 4}\,$x\not= 2$

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic55.tex}
\end{center}

\textcircled{\small 2}

2s$x^2-3x-10>0$B\\
\textcircled{\small 1}\,$-5<x<2$~~~
\textcircled{\small 2}\,$x<-5,2<x$\\
\textcircled{\small 3}\,$-2<x<5$~~~
\textcircled{\small 4}\,$x<-2,5<x$

process
$(x+2)(x-5)>0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic59.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-3x-10 < 0$B\\
\textcircled{\small 1}\,$-5<x<2$~~~
\textcircled{\small 2}\,$x<-5,2<x$\\
\textcircled{\small 3}\,$-2<x<5$~~~
\textcircled{\small 4}\,$x<-2,5<x$

process
$(x+2)(x-5)<0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic59.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-x-12 \geqq 0$B\\
\textcircled{\small 1}\,$-4 \leqq x \leqq 3$~~~
\textcircled{\small 2}\,$x \leqq -4, 3 \leqq x$\\
\textcircled{\small 3}\,$-3 \leqq x \leqq 4$~~~
\textcircled{\small 4}\,$x \leqq  -3,4  \leqq x$

process
$(x+3)(x-4) \geqq 0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic62.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-x-12 \leqq 0$B\\
\textcircled{\small 1}\,$-4 \leqq x \leqq 3$~~~
\textcircled{\small 2}\,$x \leqq -4, 3 \leqq x$\\
\textcircled{\small 3}\,$-3 \leqq x \leqq 4$~~~
\textcircled{\small 4}\,$x \leqq  -3,4  \leqq x$

process
$(x+3)(x-4) \leqq 0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic62.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-4x+3 < 0$B\\
\textcircled{\small 1}\,$-3<x<-1$~~~
\textcircled{\small 2}\,$x<-3,-1<x$\\
\textcircled{\small 3}\,$1<x<3$~~~
\textcircled{\small 4}\,$x<1,3<x$

process
$(x-1)(x-3)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic108.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-4x+3 > 0$B\\
\textcircled{\small 1}\,$-3<x<-1$~~~
\textcircled{\small 2}\,$x<-3,-1<x$\\
\textcircled{\small 3}\,$1<x<3$~~~
\textcircled{\small 4}\,$x<1,3<x$

process
$(x-1)(x-3)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic108.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-x-6 < 0$B\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$\\
\textcircled{\small 5}\,$-3<x<-2$

process
$(x+2)(x-3)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic109.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-x-6 > 0$B\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$\\
\textcircled{\small 5}\,$-3<x<-2$

process
$(x+2)(x-3)>0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic109.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-6x+5< 0$B\\
\textcircled{\small 1}\,$-5<x<-1$~~~
\textcircled{\small 2}\,$x<-5,-1<x$\\
\textcircled{\small 3}\,$1<x<5$~~~
\textcircled{\small 4}\,$x<1,5<x$

process
$(x-1)(x-5)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic110.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-6x+5 > 0$B\\
\textcircled{\small 1}\,$-5<x<-1$~~~
\textcircled{\small 2}\,$x<-5,-1<x$\\
\textcircled{\small 3}\,$1<x<5$~~~
\textcircled{\small 4}\,$x<1,5<x$

process
$(x-1)(x-5)>0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic110.tex}
\end{center}

\textcircled{\small 4}

2s$x^2+x-6< 0$B\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$

process
$(x+3)(x-2)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic47.tex}
\end{center}

\textcircled{\small 1}

2s$x^2+x-6>0$B\\
\textcircled{\small 1}\,$-3<x<2$~~~
\textcircled{\small 2}\,$x<-3,2<x$\\
\textcircled{\small 3}\,$-2<x<3$~~~
\textcircled{\small 4}\,$x<-2,3<x$

process
$(x+3)(x-2)>0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic47.tex}
\end{center}

\textcircled{\small 2}

2s
$(x-5)(x-7)\geqq 0$
ƁC̉\fbox{\bf\,E\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$5\leqq x\leqq 7$~~~
\textcircled{\small 2}\,$x\leqq 5,\,7\leqq x$\\
\textcircled{\small 3}\,$-7\leqq x\leqq -5$~~~
\textcircled{\small 4}\,$x\leqq -7,\,-5\leqq x$

\textcircled{\small 2}

2s
$(x+1)(x-2)< 0$
ƁC̉\fbox{\bf\,L\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$-1< x < 2$~~~
\textcircled{\small 2}\,$x< -1,\,2< x$\\
\textcircled{\small 3}\,$-2< x< 1$~~~
\textcircled{\small 4}\,$x< -2,\,1< x$

\textcircled{\small 1}

2s$-x^2+x+2>0$ƁC
̉\fbox{\bf\,J\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB
C}2֐$y=-x^2+x+2$̃OtłB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$-1< x < 2$~~~
\textcircled{\small 2}\,$x< -1,\,2< x$\\
\textcircled{\small 3}\,$-2< x< 1$~~~
\textcircled{\small 4}\,$x< -2,\,1< x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2602m0403.tex}
\end{center}

\textcircled{\small 1}

2s$(x-1)(x-2)\leqq 0$ƁC
̉\fbox{\bf\,J\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\textcircled{\small 1}\,$x\leqq -2,\,-1\leqq x$~~~
\textcircled{\small 2}\,$-2\leqq x \leqq -1$\\
\textcircled{\small 3}\,$x\leqq 1,\,2\leqq x$~~~
\textcircled{\small 4}\,$1\leqq x \leqq 2$

\textcircled{\small 4}

2sȂB\\
$x^2-x-12 < 0$\\
\textcircled{\small 1}\,$x\leqq -4,\,3\leqq x$~~~
\textcircled{\small 2}\,$-4\leqq x \leqq 3$\\
\textcircled{\small 3}\,$x\leqq -3,\,4\leqq x$~~~
\textcircled{\small 4}\,$-3\leqq x \leqq 4$

process
$x^2-x-12<0$\\
$(x+3)(x-4)<0$

\textcircled{\small 4}

2sȂB\\
$x^2+x+12 > 0$\\
\textcircled{\small 1}\,$x\leqq -4,\,3\leqq x$~~~
\textcircled{\small 2}\,$-4\leqq x \leqq 3$\\
\textcircled{\small 3}\,$x\leqq -3,\,4\leqq x$~~~
\textcircled{\small 4}\,$-3\leqq x \leqq 4$

process
$x^2-x-12>0$\\
$(x+3)(x-4)>0$

\textcircled{\small 3}

2s$x^2-7x+10 > 0$B\\
\textcircled{\small 1}\,$-5< x < -2$~~~
\textcircled{\small 2}\,$x< -5,\,-2< x$\\
\textcircled{\small 3}\,$2< x< 5$~~~
\textcircled{\small 4}\,$x< 2,\,5< x$

process
$(x-2)(x-5)>0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic111.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-7x+10 < 0$B\\
\textcircled{\small 1}\,$-5< x < -2$~~~
\textcircled{\small 2}\,$x< -5,\,-2< x$\\
\textcircled{\small 3}\,$2< x< 5$~~~
\textcircled{\small 4}\,$x< 2,\,5< x$

process
$(x-2)(x-5)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic111.tex}
\end{center}

\textcircled{\small 3}

2s$x^2+2x-15 < 0$B\\
\textcircled{\small 1}\,$-5< x < 3$~~~
\textcircled{\small 2}\,$x< -5,\,3< x$\\
\textcircled{\small 3}\,$-3< x< 5$~~~
\textcircled{\small 4}\,$x< -3,\,5< x$

process
$(x+5)(x-3) < 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic113.tex}
\end{center}

\textcircled{\small 1}

2s$x^2+2x-15 > 0$B\\
\textcircled{\small 1}\,$-5< x < 3$~~~
\textcircled{\small 2}\,$x< -5,\,3< x$\\
\textcircled{\small 3}\,$-3< x< 5$~~~
\textcircled{\small 4}\,$x< -3,\,5< x$

process
$(x+5)(x-3) > 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic113.tex}
\end{center}

\textcircled{\small 2}

2s$x^2-8x+15<0$B\\
\textcircled{\small 1}\,$-5< x < -3$~~~
\textcircled{\small 2}\,$x< -5,\,-3< x$\\
\textcircled{\small 3}\,$3< x< 5$~~~
\textcircled{\small 4}\,$x< 3,\,5< x$

process
$(x-3)(x-5)<0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic114.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-8x+15>0$B\\
\textcircled{\small 1}\,$-5< x < -3$~~~
\textcircled{\small 2}\,$x< -5,\,-3< x$\\
\textcircled{\small 3}\,$3< x< 5$~~~
\textcircled{\small 4}\,$x< 3,\,5< x$

process
$(x-3)(x-5)>0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic114.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-6x+8 > 0$B\\
\textcircled{\small 1}\,$-4< x < -2$~~~
\textcircled{\small 2}\,$x< -4,\,-2< x$\\
\textcircled{\small 3}\,$2< x< 4$~~~
\textcircled{\small 4}\,$x< 2,\,4< x$

process
$(x-2)(x-4)>0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic115.tex}
\end{center}

\textcircled{\small 4}

2s$x^2-6x+8 < 0$B\\
\textcircled{\small 1}\,$-4< x < -2$~~~
\textcircled{\small 2}\,$x< -4,\,-2< x$\\
\textcircled{\small 3}\,$2< x< 4$~~~
\textcircled{\small 4}\,$x< 2,\,4< x$

process
$(x-2)(x-4)<0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic115.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-7x+6 < 0$B\\
\textcircled{\small 1}\,$-6< x < -1$~~~
\textcircled{\small 2}\,$x< -6,\,-1< x$\\
\textcircled{\small 3}\,$1< x< 6$~~~
\textcircled{\small 4}\,$x< 1,\,6< x$

process
$(x-1)(x-6) < 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic116.tex}
\end{center}

\textcircled{\small 3}

2s$x^2-7x+6 > 0$B\\
\textcircled{\small 1}\,$-6< x < -1$~~~
\textcircled{\small 2}\,$x< -6,\,-1< x$\\
\textcircled{\small 3}\,$1< x< 6$~~~
\textcircled{\small 4}\,$x< 1,\,6< x$

process
$(x-1)(x-6) > 0$ \\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic116.tex}
\end{center}

\textcircled{\small 4}

2sȂB\\
$x^2+x-30 \leqq 0$\\
\textcircled{\small 1}\,$-6\leqq x\leqq 5$~~~
\textcircled{\small 2}\,$x\leqq -6,\, 5\leqq x$\\
\textcircled{\small 3}\,$-5\leqq x\leqq 6$~~~
\textcircled{\small 4}\,$x\leqq -5,\, 6\leqq x$

process
$(x+6)(x-5) \leqq 0$

\textcircled{\small 1}

2sȂB\\
$x^2+x-30 \geqq 0$\\
\textcircled{\small 1}\,$-6\leqq x\leqq 5$~~~
\textcircled{\small 2}\,$x\leqq -6,\, 5\leqq x$\\
\textcircled{\small 3}\,$-5\leqq x\leqq 6$~~~
\textcircled{\small 4}\,$x\leqq -5,\, 6\leqq x$

process
$(x+6)(x-5) \geqq 0$

\textcircled{\small 2}

2sȂB\\
$x^2+3x-10 \leqq 0$\\
\textcircled{\small 1}\,$-5\leqq x\leqq 2$~~~
\textcircled{\small 2}\,$x\leqq -5,\, 2\leqq x$\\
\textcircled{\small 3}\,$-2\leqq x\leqq 5$~~~
\textcircled{\small 4}\,$x\leqq -2,\, 5\leqq x$

process
$(x+5)(x-2) \leqq 0$

\textcircled{\small 1}

2sȂB\\
$x^2+3x-10 \geqq 0$\\
\textcircled{\small 1}\,$-5\leqq x\leqq 2$~~~
\textcircled{\small 2}\,$x\leqq -5,\, 2\leqq x$\\
\textcircled{\small 3}\,$-2\leqq x\leqq 5$~~~
\textcircled{\small 4}\,$x\leqq -2,\, 5\leqq x$

process
$(x+5)(x-2) \geqq 0$

\textcircled{\small 2}

2sȂB\\
$x^2+3x \leqq 0$\\
\textcircled{\small 1}\,$-3\leqq x\leqq 0$~~~
\textcircled{\small 2}\,$x\leqq -3,\, 0\leqq x$\\
\textcircled{\small 3}\,$0\leqq x\leqq 3$~~~
\textcircled{\small 4}\,$x\leqq 0,\, 3\leqq x$

process
$x(x+3) \leqq 0$

\textcircled{\small 1}

2sȂB\\
$x^2+3x \geqq 0$\\
\textcircled{\small 1}\,$-3\leqq x\leqq 0$~~~
\textcircled{\small 2}\,$x\leqq -3,\, 0\leqq x$\\
\textcircled{\small 3}\,$0\leqq x\leqq 3$~~~
\textcircled{\small 4}\,$x\leqq 0,\, 3\leqq x$

process
$x(x+3) \geqq 0$

\textcircled{\small 2}

2s$x^2+2x-3 \geqq 0$B\\
\textcircled{\small 1}\,$-3\leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x\leqq -3,\,1\leqq x$\\
\textcircled{\small 3}\,$-1\leqq x\leqq 3$~~~
\textcircled{\small 4}\,$x\leqq -1,\,3\leqq x$

process
$(x+3)(x-1)\geqq0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic112.tex}
\end{center}

\textcircled{\small 2}

2s$x^2+2x-3 \leqq 0$B\\
\textcircled{\small 1}\,$-3\leqq x \leqq 1$~~~
\textcircled{\small 2}\,$x\leqq -3,\,1\leqq x$\\
\textcircled{\small 3}\,$-1\leqq x\leqq 3$~~~
\textcircled{\small 4}\,$x\leqq -1,\,3\leqq x$

process
$(x+3)(x-1)\leqq 0$\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic112.tex}
\end{center}

\textcircled{\small 1}

2s$-(x-3)(x-7)\leqq 0$ƁC
̉\fbox{\bf\,J\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB\\
\hspace{8pt} 
C}́C2֐$y=-(x-3)(x-7)$̃OtłB\\
\textcircled{\small 1}\,$-7\leqq x\leqq -3$~~~
\textcircled{\small 2}\,$x\leqq -7,\, -3\leqq x$\\
\textcircled{\small 3}\,$3\leqq x\leqq 7$~~~
\textcircled{\small 4}\,$x\leqq 3,\, 7\leqq x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m0401.tex}
\end{center}

\textcircled{\small 4}

2s
$(3x-1)(2x-1)\geqq 0$̉\fbox{\,\bf I \,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$\displaystyle\frac{1}{3} \leqq x \leqq \displaystyle\frac{1}{2}$~~~
\textcircled{\small 2}\,$-\displaystyle\frac{1}{2} \leqq x \leqq -\displaystyle\frac{1}{3}$\\
\textcircled{\small 3}\,$x \leqq -\displaystyle\frac{1}{2} ,-\displaystyle\frac{1}{3} \leqq x$~~~
\textcircled{\small 4}\,$x\leqq \displaystyle\frac{1}{3} , \displaystyle\frac{1}{2}\leqq x$

\textcircled{\small 4}

\textcircled{\small 1}$\sim$\textcircled{\small 4}2s̒ŁC
ׂ̉Ă̎ł̂\fbox{\,\bf P\,}łB
̂IׁB
C}́C2֐$y=(x-1)^2$̃OtłB\\
\textcircled{\small 1}\,$y=(x-1)^2>0$~~~
\textcircled{\small 2}\,$y=(x-1)^2<0$\\
\textcircled{\small 3}\,$y=(x-1)^2\geqq 0$~~~
\textcircled{\small 4}\,$y=(x-1)^2\leqq 0$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2802401.tex}
\end{center}

\textcircled{\small 3}

2s
$(x-3)(x-6)> 0$
ƁC̉\fbox{\bf\,N\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$3< x < 6$~~~
\textcircled{\small 2}\,$x< 3,\,6< x$\\
\textcircled{\small 3}\,$-6< x< -3$~~~
\textcircled{\small 4}\,$x< -6,\,-3< x$

\textcircled{\small 2}

2s$x^2-x<0$ƁC
̉\fbox{\bf\,L\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB
C}2֐$y=x^2-x$̃OtłB
\\
\textcircled{\small 1}\,$0< x < 1$~~~
\textcircled{\small 2}\,$x< 0,\,1< x$\\
\textcircled{\small 3}\,$-1< x< 0$~~~
\textcircled{\small 4}\,$x< -1,\,0< x$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h3002401.tex}
\end{center}

\textcircled{\small 1}

2s
$x^2-2x+1\geqq 0$
ƁC̉\fbox{\bf\hspace{4pt} J \hspace{4pt}}B
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB
C}́C2֐
$y=x^2-2x+1$̃OtłB\\
\textcircled{\small 1}\,ׂĂ̎~~~
\textcircled{\small 2}\,1ȊÔׂĂ̎\\
\textcircled{\small 3}\,$x=1$~~~
\textcircled{\small 4}\,Ȃ
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pich31401.tex}
\end{center}

\textcircled{\small 1}

2s
\fbox{\hspace{4pt} 5 \hspace{4pt}}łB\\
$x^2-2x-8<0$\\
\textcircled{\small 1}\,$x<-4,2<x$~~~
\textcircled{\small 2}\,$-4<x<2$\\
\textcircled{\small 3}\,$x<2,4<x$~~~
\textcircled{\small 4}\,$-2<x<4$\\
\textcircled{\small 5}\,$-4<x<-2$

process
$(x+2)(x-4)< 0$\\
$-2<x<4$

\textcircled{\small 4}

2s
\fbox{\hspace{4pt} 5 \hspace{4pt}}łB\\
$x^2-3x-40<0$\\
\textcircled{\small 1}\,
$x<-8,5<x$~~~
\textcircled{\small 2}\,
$x<5,8<x$\\
\textcircled{\small 3}\,
$x<-5,8<x$~~~
\textcircled{\small 4}\,
$-5<x<8$\\
\textcircled{\small 5}\,
$-8<x<5$

process
$(x+5)(x-8)< 0$\\
$-5<x<8$

\textcircled{\small 4}










[Level7]
p
$y=x^2-6x+5$B̖₢ɓȂB\\
(I)\,̕$x$Ƃ̌_ACBƂƂCAB̒߂ȂB\\
(I\hspace{-.1em}I)\,_PƂƂC$\triangle$PAB̖ʐς߂ȂB

process
(I)\,$0=x^2-6x+5$ƁC$0=(x-1)(x-5)$ƈł邩C
$x=1,\,5$\\
āCAB̒$5-1=4$\\
(I\hspace{-.1em}I)\,
$y=x^2-6x$ƁC$y=x(x-6)$ƂȂC$y=x^2-6x$$x$$x=0,\,6$ŌBāC$y=x^2-6x+5$̒_
$(3,\,-4)$B\\
߂ʐς$\displaystyle\frac{1}{2} \times 4 \times 4=8$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic135.tex}
\end{center}

8

$y=x^2-2x-3-k$$x$ƋL_ƂC萔$k$̒l͈̔͂߂ȂB

process
$0=x^2-2x-3-k$\\
$\Longleftrightarrow k=x^2-2x-3$\\
$y=x^2-2x-3$$y=k$L_悤$k$͈̔͂߂΂悢B\\
$y=x^2-2x-3$̒_$(1,\,-4)$
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/pic136.tex}
\end{center}

$k\geqq -4$


$y=x^2-4x+k$$x$ƋL_ȂƂC
萔$k$̒l͈̔͂߂ȂB

process
$x^2-4x+k=0$̔ʎ$D$ƂƁC\\
$D=(-4)^2-4k=16-4k<0$ł΂悢B
āC$k>4$

$k>4$


$y=x^2+8x+17+k$$x$ƋL_ƂC
萔$k$̒l͈̔͂߂ȂB

process
$x^2+8x+17+k=0$̔ʎ$D$ƂƁC$b'=4$\\
$\displaystyle\frac{D}{4}=4^2-(17+k)=16-k\geqq 0$ł΂悢B
āC$k\leqq -1$

$k\leqq -1$


$y=x^2+4x+1+k$$x$ƋL_ȂƂC
萔$k$̒l͈̔͂߂ȂB

process
$x^2+4x+1+k=0$̔ʎ$D$ƂƁC$b'=2$\\
$\displaystyle\frac{D}{4}=2^2-(1+k)=3-k< 0$ł΂悢B
āC$k> 3$

$k> 3$











[Level8]
Qϐ2֐
$x+y=2$̂ƂC֐$x^2+y^2$̍ől߂B

process
$x+y=2$C$y=-x+2$ł邩C$x^2+y^2$ɑāC\\
$x^2+(-x+2)^2=2x^2-4x+4=2(x-1)^2+2$

ŏlF2i$x=1$C$y=1$̂Ƃj












[EOF]
ڂ

2s$x^2+2x-3>0$ƁC
̉\fbox{\bf\,N\,}łB
\textcircled{\small 1}$\sim$\textcircled{\small 4}
̂琳̂IׁB
C}2֐$y=x^2+2x-3$̃OtłB
\textcircled{\small 1}$\sim$\textcircled{\small 4}̂琳̂IׁB\\
\textcircled{\small 1}\,$-3< x < 1$~~~
\textcircled{\small 2}\,$x< -3,\,1< x$\\
\textcircled{\small 3}\,$-1< x< 3$~~~
\textcircled{\small 4}\,$x< -1,\,3< x$~~~
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/212_2jifunction/h2801m0304.tex}
\end{center}

\textcircled{\small 2}


% t@C̍Ōɂ
