【数学I】例題3.1.4：2次関数のグラフ1（One More）★

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% 例題I3.1.4：2次関数のグラフ1（One More）★
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次の2次関数のグラフは，2次関数$y=2x^2$のグラフをそれぞれどのように平行移動したものか．また，それぞれのグラフをかき，その軸と頂点を求めよ．

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

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

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

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% 例題I3.1.4の解答（One More）★
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(1) $y$軸方向に$-1$だけ平行移動したものであり，グラフは右の図のようになる．

軸は$y$軸（直線$x=0$）

頂点は点$(0,-1)$

(2) $x$軸方向に$-1$だけ平行移動したものであり，グラフは右の図のようになる．

軸は直線$x=-1$

頂点は点$(-1,0)$

(3) $x$軸方向に$1$，$y$軸方向に1だけ平行移動したものであり，グラフは右の図のようになる．

軸は直線$x=1$

頂点は点$(1,1)$

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% 問題I3.1.4：2次関数のグラフ1（One More）★
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次の2次関数のグラフは，2次関数$y=-x^2$のグラフをそれぞれどのように平行移動したものか．また，それぞれのグラフをかき，その軸と頂点を求めよ．

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

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

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

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(1) $y$軸方向に$3$だけ平行移動したものであり，グラフは右の図のようになる．

軸は$y$軸（直線$x=0$）

頂点は点$(0,3)$

(2) $x$軸方向に$2$だけ平行移動したものであり，グラフは右の図のようになる．

軸は直線$x=2$

頂点は点$(2,0)$

(3) $x$軸方向に$-1$，$y$軸方向に$-4$だけ平行移動したものであり，グラフは右の図のようになる．

軸は直線$x=-1$

頂点は点$(-1,-4)$

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% 例題I3.1.4の側注：2次関数のグラフ1（One More）★
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% 側注1
$y=a(x-p)^2+q$において，$p=0$の形であるから，$y$軸方向にのみ平行移動する．

% 側注2
$y=a(x-p)^2+q$において，$q=0$の形であるから，$x$軸方向にのみ平行移動する．

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% 問題I3.1.4の側注：2次関数のグラフ1（One More）★
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% 側注1
$y=a(x-p)^2+q$において，$p=0$の形であるから，$y$軸方向にのみ平行移動する．

% 側注2
$y=a(x-p)^2+q$において，$q=0$の形であるから，$x$軸方向にのみ平行移動する．

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% 例題I3.1.4の考え方・補足など：2次関数のグラフ1（One More）★
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% 考え方
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2次関数$y=a(x-p)^2+q$のグラフは，$y=ax^2$のグラフを$x$軸方向に$p$，$y$軸方向に$q$だけ平行移動した放物線で，

軸は直線$x=p$，頂点は点$(p,q)$

また，$y=a(x-p)^2+q$のグラフは，頂点$(p,q)$を原点と見て，$y=ax^2$のグラフをかけばよい．

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グラフをかくときは，頂点の座標と頂点以外の通る1点を示す．通る1点は，$y$軸との交点など（$x=0$を代入して求められる），求めやすい点を示すのがよい．

例：2次関数$y=2x^2$をかくときは，$y$軸との交点が原点であり，頂点と一致する．通る1点として例えば，求めやすい点$(1,2)$を示す．