Licensing information

Presentation on theme: "Licensing information"— Presentation transcript:

1 Licensing information
Users should treat this material as a working draft. This material can be used in its current form, customized, redistributed and/or printed or displayed by the user. The author(s) request feedback on all materials so that they can be continually improved and updated. This material is licensed under the Creative Commons Attribution license: ( Author: Kevin Hall Wording for the legal statement above is adapted from the legal statement for Trigonometry, published in 2009 by The CK-12 Foundation:

2 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?

3 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？

4 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

5 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

6 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

7 ３０人なら？ X人なら？

8 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

9 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

10 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

11 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

12 ２次関数の形 2ndの差異が定数の場合、グラフの形は？

13 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

14 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

15 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