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[Title]
211_1jifunction

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% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
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% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB


[Level1]
1֐̃Ot
_ʂCω̊4ł钼̎߂B

$y=4x$

X$-3$ŐؕЂ5ł钼̎߂B

$y=-3x+5$

X$-2$œ_$(2,-1)$ʂ钼̎߂B

process
$y=-2x+b$_$(2,-1)$ʂ邩C\\
$-1=-2 \times 2 +b$C$b=3$

$y=-2x+3$

$x$Ƃ̌_$(3,\,0)$C$y$Ƃ̌_$(0,\,-4)$
ł钼̎߂B

process
$y=ax-4$$(3,0)$$0=3a-4$C$a=\displaystyle\frac{4}{3}$C
$y=\displaystyle\frac{4}{3} x-4$\\
iʉj$\displaystyle\frac{x}{3}+\displaystyle\frac{y}{-4}=1$C
$4x-3y=12$

$y=\displaystyle\frac{4}{3} x-4$܂$4x-3y=12$

$(-1,\,-8),(3,\,4)$ʂ钼̎߂B

process
X$\displaystyle\frac{4-(-8)}{3-(-1)}=\displaystyle\frac{12}{4}=3$C\\
$y=3x+b$_$(3,\,4)$ʂ邩C
$4=3 \times 3+b$C$b=-5$

$y=3x-5$

ω̊2ŁC$x=1$̂Ƃ$y=-1$ƂȂ1֐̎߂ȂB

process
$y=2x+b$_$(1,\,-1)$ʂ邩C$-1=2\times 1+b$C$b=-3$

$y=2x-3$

$x=-6$̂Ƃ$y=1$C$x=3$̂Ƃ$y=7$ł1֐̎߂ȂB

process
X$=\displaystyle\frac{7-1}{3-(-6)}=\displaystyle\frac{2}{3}$B
$y=\displaystyle\frac{2}{3}(x-3)+7$

$y=\displaystyle\frac{2}{3} x+5$

1֐$y=ax+4$̃Ot2_$(2,\,3)$C$(4,\,b)$ʂƂC
$aCb$̒lꂼꋁ߂ȂB

process
1֐$y=ax+4$̃Ot2_$(2,\,3)$ʂ邩C$3=a\times 2+4$C
$a=-\displaystyle\frac{1}{2}$B1֐$y=-\displaystyle\frac{1}{2}x+4$B

$a=-\displaystyle\frac{1}{2},\,b=2$

2_$(-1,\,-5)$C$(1,\,3)$ʂ钼̎߂B

process
X$\displaystyle\frac{3-(-5)}{1-(-1)}=4$B
߂钼̎$y=4(x-1)+3=4x-1$

$y=4x-1$

$y=\displaystyle\frac{1}{2} x+b$C2̒$y=3x-1$C$y=-x+3$̌_ʂƂC$b$̒l߂B

process
2̌_$(1,\,2)$łB
$y=\displaystyle\frac{1}{2} x+b$̌_ʂ邩C$2=\displaystyle\frac{1}{2} \times 1+b$
C$b=\displaystyle\frac{3}{2}$

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

OťX3ŁC_$(-2,\,4)$ʂ1֐̎߂B

process
$y=3(x+2)+4=3x+10$

$y=3x+10$

OťX2ŁC_$(3,\,-4)$ʂ1֐̎߂B

process
$y=2(x-3)-4=2x-10$

$y=2x-10$

OťX3ŁC_$(2,\,-4)$ʂ1֐̎߂B

process
$y=3(x-2)-4=3x-10$

$y=3x-10$

2_$(-3,\,-7)$C$(-2,\,8)$ʂ1֐̎߂B

process
$y=\displaystyle\frac{8-(-7)}{-2-(-3)}(x+3)-7=15(x+3)+7=15x+38$

$y=15x+38$

$y=-4x$ɕsŁC_$(-3,\,6)$ʂ1֐̎߂B

process
$y=-4\{x-(-3)\}+6=-4x-6$

$y=-4x-6$

_Ɠ_$(-4,\,-6)$ʂ1֐̎߂B

process
$y=\displaystyle\frac{0-(-6)}{0-(-4)}x=\displaystyle\frac{3}{2}x$

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

$x=1$̂Ƃ$y=1$C$x=5$̂Ƃ$y=4$ƂȂ1֐̎߂B

process
$y=\displaystyle\frac{4-1}{5-1}(x-1)+1$

$y=\displaystyle\frac{3}{4} x+\displaystyle\frac{1}{4}$

2_$(-1,\,-5)$C$(1,\,3)$ʂ钼̎߂B

process
X$\displaystyle\frac{3-(-5)}{1-(-1)}=4$B
߂钼̎$y=4(x-1)+3=4x-1$

$y=4x-1$

$y=\displaystyle\frac{1}{2} x+b$C2̒$y=3x-1$C$y=-x+3$̌_ʂƂC$b$̒l߂B

process
2̌_$(1,\,2)$łB
$y=\displaystyle\frac{1}{2} x+b$̌_ʂ邩C$2=\displaystyle\frac{1}{2} \times 1+b$
C$b=\displaystyle\frac{3}{2}$

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







[Level2]
1֐̕sړ
$y=2x$$x$$-2$C$y$$3$sړƂ̒̎߂B

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

$y=2x+7$

$y=2x+5$̃Ot$x$6C$y$$-3$
sړOt\B

process
$y=2x+5$$x$$x-6$C$y$$y+3$΂悢B\\
$y+3=2(x-6)+5$C$y=2x-10$

$y=2x-10$

$y=2x+4$$x$$3$sړ߂B

process
$y=2(x-3)+4=2x-2$

$y=2x-2$

$y=2x+4$$y$$-6$sړ߂B

process
$y=2x+4-6=2x-2$

$y=2x-2$

$y=2x+4$$x$$-4$C$y$$2$sړ߂B

process
$y=2(x+4)+4+2=2x+14$

$y=2x+14$

[Level3]
1֐̑Ώ̈ړ
$y=3x-6$$x$ɂđΏ̈ړ߂B

process
$y$$-y$ɒuƁC$-y=3x-6$C$y=-3x+6$

$y=-3x+6$

$y=3x-6$$y$ɂđΏ̈ړ߂B

process
$x$$-x$ɒuƁC$y=3(-x)-6$C$y=-3x-6$

$y=-3x-6$

$y=3x-6$_ɂđΏ̈ړ߂B

process
$x$$-x$ɁC$y$$-y$ɒuƁC$-y=3(-x)-6$C$y=3x+6$

$y=3x+6$







[Level4]
1֐̉p
$x$Ƃ̌_$x$W5C$y=3x+1$Ƃ̌_$x$W1ł钼̎߂ȂB

process
$x$Ƃ̌_$x$W5C߂钼̎$y=a(x-5)$ƂB
ƒ$y=3x+1$Ƃ̌_$x$W1ł邱ƂC
$a(x-5)=3x+1$̉$x=1$ƂȂ邩C$a(1-5)=3\times 1+1$C$a=-1$
߂钼̎$y=-(x-5)=-x+5$

$y=-(x-5)=-x+5$

}̂悤ɁC2̒$l$C$m$C_PŌĂ܂B
_P̍W߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/211_1jifunction/pic01.tex}
\end{center}

process
$l:\,y=2x+4$\\
$m:\,y=-\displaystyle\frac{2}{3}x+2$\\
C$\left( -\displaystyle\frac{3}{4},\,\displaystyle\frac{5}{2} \right)$

$\left( -\displaystyle\frac{3}{4},\,\displaystyle\frac{5}{2} \right)$

}̂悤ɁC2_A$(1,\,4)$CB$(3,\,1)$܂B
$y$ɓ_PƂCAP+PB̒l܂B
AP$+$PB̒łZȂƂC_P̍W߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/211_1jifunction/pic02.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/211_1jifunction/pic02p.tex}
\end{center}
_B$y$ɂđΏ̂ȓ_B$'$̍W$(-3,\,1)$B
AB$'$̎$y=\displaystyle\frac{3}{4}(x-1)+4=\displaystyle\frac{3}{4} x+\displaystyle\frac{13}{4}$B
̒$y$Ƃ̌_$\left( 0,\,\displaystyle\frac{13}{4} \right)$

$\left( 0,\,\displaystyle\frac{13}{4} \right)$

}̂悤ɁC֐$y=ax$̃Ot2_A$(3,\,8)$CB$(9,\,0)$܂B
֐$y=ax$̃Ot$\triangle$AOB̖ʐς2ƂC$a$̒l߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/211_1jifunction/pic03.tex}
\end{center}

process
AB̒_̍ẂC$\left( \displaystyle\frac{3+9}{2},\displaystyle\frac{8+0}{2} \right)=(6,\,4)$łB
߂Ot͌_OƐAB̒_ʂ钼łB̂ƂC$a=\displaystyle\frac{4}{6}=\displaystyle\frac{2}{3}$

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

$y=ax+8$C2̒$2x-5y=11$C$3x+2y=7$̌_ʂƂC
$a$̒l߂B

process
$\left\{ 
\begin{array}{l}
 2x-5y=11\cdots \textcircled{\small 1} \\
 3x+2y=7 \cdots \textcircled{\small 2}
\end{array} \right.$
$\left\{ 
\begin{array}{l}
 \textcircled{\small 1} \times -3\,:\,-6x+15y=-33\cdots \textcircled{\small 1}' \\
 \textcircled{\small 2} \times 2\,:\,6x+4y=14 \cdots \textcircled{\small 2}'
\end{array} \right.$
\\ \textcircled{\small 1}$'+$\textcircled{\small 2}$'$C$y=-1$C$x=3$ \\
$y=ax+8$$(3,\,-1)$ʂ邩C$-1=a \times 3+8$C$a=-3$

$-3$

$(-2,\,-5)$ʂC$x-3y-3=0$ɐȒ̕߂B

process
$x-3y-3=0$͕ό`ƁC$y=\displaystyle\frac{1}{3}x-1$ł邩C
̒̌X$\displaystyle\frac{1}{3}$B
߂钼̌X$m$ƂƁCӂC\\
$m\cdot\displaystyle\frac{1}{3}=-1$C$m=-3$B
āC߂钼\\
$y=-3(x+2)-5=-3x-11$

$y=-3x-11\,i3x+y+11=0j$






[Level5]


[Level6]


[Level7]



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