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Objectives: How do the concepts from Ch. 4 relate to the real world? What are 1st and 2nd differences, and how can you use them to identify different families of functions?
線形と非線形 言葉の意味は？ 実世界でグラフが直線になるのは？
例: グラフが直線になると どうして分かる？ x y 1 +3 1 4 +3 2 7 +3 3 10 +2 +6 5 16 y x +6 20 グラフが直線になると どうして分かる？ 15 +6 x y 1 10 +3 +2 1 4 +3 +3 2 7 5 +3 +3 3 10 +2 +6 +3 5 16 +1 1 2 3 4 5 6 7 8 9 10 x
非線形関数 変化の割合が一定ではないので、直線にはならない
Facebook の例 みんながFacebookで自分も含めて友達のページに２回ずつ書き込みをしたら、全体ではどれだけになる？ ステップ 1: 1 人 2 人 3 人 __ 回 書き込み 2 ４回 4 回 ６回 ６回 ６回 計： ４×２ 計： ６×３ 計: 2
３０人なら？ X人なら？
２次関数 上の例の関数 f(x) = 2x2 元になるのは f(x) = x2 ２次関数の例: f(x) = (x – 4)2 f(x) = x2 + 3x + 2 x の項が x2
２次関数はどれ？? f(x) = x2 f(x) = 9 – x2 + 5x f(x) = x2 + 8x3 Yes. Yes x の項はx3.したがってNo
Facebook の例で変化率は？ 1 人 2 人 3 人 __ 回 書き込み 2 ４回 4 回 ６回 ６回 ６回 計： ４×２ 計： ６×３ 計: 2 ?? + 6 ?? + 10 グラフは直線？理由は
1st と 2nd の差異 y = 3x + 1 y = x2 x y x y 1st Diff. 1st Diff. 2nd Diff. -3 -8 -3 9 -2 -5 -2 4 -1 -2 -1 1 1 1 4 1 1 2 7 2 4 3 10 3 9
２次関数の形 2ndの差異が定数の場合、グラフの形は？
1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 +1 +2 +1 +1 +1 +1
1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 -2 +2 +1 -1 +1 +1 +1 +0 +1 +1
1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 -2 +2 +1 -1 +1 +1 +1 +0 +1 +1 2nd Differences: 1st Differences: 1, 1, 1, 1, 1, 1 1st Differences: -2, -1, 0, 1, 2, 3