| English | 意味 |
|---|---|
| quadratic function/kwɒˈdrætɪk ˈfʌŋkʃn/ | 二次関数 |
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ | 一次関数 |
| constant/ˈkɒnstənt/ | 定数 |
| slope/sləʊp/ | 傾き |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ | 平均変化率 |
Two kinds of climb
- An escalator lifts you at a steady pace: every second gains the same height.
- A jet on the runway is different: each second it gains more speed than the last.
- Both are "increasing", but the way they change is completely different.
- Linear and quadratic functions are the first two members of this story.
A linear function: constant rate
- A linear function 线性函数 graphs as a straight line.
- Its slope 斜率 is the same everywhere, so its average rate of change 平均变化率 is the same over any interval.
- Over $[0, 2]$ or $[10, 12]$ or any other stretch — you always get the identical number.
- That single unchanging rate is what "linear" really means.
一次関数の任意の区間における平均変化率は……
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直線には固定された傾きが一つだけあるため、平均変化率は あらゆる 区間で同じです——これが一次関数である理由です。
A quadratic function: a changing rate
- A quadratic function 二次函数 graphs as a curved parabola.
- Because it bends, its slope is different at different places — the rate of change keeps changing.
- Near the vertex it is nearly flat; far from the vertex it is steep.
- So its average rate of change depends on which interval you pick.
スライダーを操作すると、放物線が直線になります。
y = ax² + bx + c
a = 0 に設定すると直線になります——その傾きは一定です。a ≠ 0 にすると曲線が湾曲するため、変化率は continuously 変化します。
二次関数において、平均変化率はすべての区間で同じです。
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放物線は湾曲するため、傾きは continuously 変化します——異なる区間は異なる平均変化率を示します。
Equal steps reveal the pattern
- Take equal steps in the input and list the outputs of $y = x^2$: $0, 1, 4, 9, 16$.
- The jumps between them — the first differences — are $1, 3, 5, 7$.
- Those jumps are not constant, but they grow by the same amount ($2$) every time.
- So a quadratic's rate of change is not fixed, yet it changes in a perfectly steady, linear way.
$y = x^2$ について、$x = 1, 2, 3$ における一階差分は $3, 5, 7$ です。各ステップで一定量だけ増加しますか？
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$5 - 3 = 2$ と $7 - 5 = 2$ 。この一定の二階差分は二次関数の指紋のようなものです。
二次関数において、$x$ での等間隔のステップは……
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一階差分自体が 一次 パターンを形成し、各ステップで同じ量だけ増加します。（毎回倍になるのは指数関数的です。）
Every average rate is a secant slope
- Whichever function you have, the average rate of change over $[a, b]$ is the slope of the secant line through its endpoints.
- For a line, every secant lies right on top of the graph — same slope every time.
- For a parabola, each secant tilts differently — that is the visible sign of a changing rate.
- Reading secant slopes is how you compare how two functions change.
$[a, b]$ における平均変化率を求める手順を正しい順序に並べ替えてください。
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まず出力を求め、次にその差を取り、最後に入力の差で除算します——これが割線の傾きです。
正しい記述をすべて select せよ。
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二次関数の変化率は 一定ではありません ——一定のペースで変化します（二階差分が一定です）。他の3つの記述は正しいです。
Do not call a quadratic's rate "constant" just because the pattern is predictable. The rate itself changes from interval to interval; it is the change in the rate that is constant. For a straight line, that change in the rate is zero.
A dropped ball falls a distance $d = 5t^2$ metres after $t$ seconds.
- Distances at $t = 0, 1, 2, 3$: $0, 5, 20, 45$ metres.
- First differences (distance each second): $5, 15, 25$ — growing by a constant $10$.
- The ball's speed rises linearly even though the distance is quadratic.
A linear function changes at one constant rate — every secant has the same slope. A quadratic function's rate of change is not constant: over equal steps its first differences grow by a fixed amount, so the rate changes steadily. The average rate of change over $[a, b]$ is always the secant slope.