Polynomial Functions and Rates of Change · 多项式函数与变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| degree/dɪˈɡriː/ | 次数 | cì shù |
| leading term/ˈliːdɪŋ tɜːm/ | 首项 | shǒu xiàng |
| polynomial function/ˌpɒlɪˈnəʊmɪəl ˈfʌŋkʃn/ | 多项式函数 | duō xiàng shì hán shù |
| local maximum/ˈləʊkl ˈmæksɪməm/ | 极大值 | jí dà zhí |
| local minimum/ˈləʊkl ˈmɪnɪməm/ | 极小值 | jí xiǎo zhí |
| point of inflection/pɔɪnt ɒv ɪnˈflekʃn/ | 拐点 | guǎi diǎn |
| concavity/kənˈkævɪti/ | 凹凸性 | āo tū xìng |
How many wiggles can a curve have?
- A line is straight; a parabola has one bend; but some curves rise and fall several times.
- What decides how many hills and valleys a graph is allowed to have?
- The answer is hidden in one number in the function's rule.
- Meet the polynomials — the functions built from powers of $x$.
一条曲线能有多少个"摆动"?
- 直线是笔直的;抛物线有一个弯;但有些曲线会上下起伏好几次。
- 是什么决定了一条图像被允许拥有多少座山峰和山谷?
- 答案藏在函数式子里的一个数中。
- 来认识多项式——由 $x$ 的幂构成的函数。
Degree and leading term
- A polynomial function 多项式函数 is a sum of terms like $3x^4$, $-2x$, and $7$.
- Its degree 次数 is the highest power of $x$ that appears.
- The leading term 首项 is the term with that highest power — it dominates the graph for large $x$.
- Degree is the single most important number: it caps how wiggly the curve can be.
次数与首项
- 多项式函数(polynomial function)是若干项之和,比如 $3x^4$、$-2x$ 和 $7$。
- 它的次数(degree)是出现的 $x$ 的最高次幂。
- 首项(leading term)是带有那个最高次幂的项——当 $x$ 很大时,它主宰整张图像。
- 次数是最重要的一个数:它给曲线能摆动的次数设了上限。
The degree of a polynomial is… · 多项式的次数是……
The degree is the highest exponent on the variable — e.g. $2x^4 - x + 7$ has degree $4$. It controls the graph's overall shape and end behavior. · 次数是变量上的最高指数——例如 $2x^4 - x + 7$ 的次数是 $4$。它决定了图像的整体形状和末端行为。
Local maxima and minima
- Follow a polynomial left to right and mark where it changes direction.
- A local maximum 极大值 is a peak: increasing turns to decreasing.
- A local minimum 极小值 is a valley: decreasing turns to increasing.
- Between these turning points the function is steadily increasing or decreasing.
极大值与极小值
- 从左到右跟踪一个多项式,标出它改变方向的地方。
- 极大值(local maximum)是一座山峰:递增转为递减。
- 极小值(local minimum)是一个山谷:递减转为递增。
- 在这些转折点之间,函数稳定地递增或递减。
Reshape a cubic and count its turning points · 重塑一个三次函数,数一数它的转折点
y = ax³ + bx² + cx + d
Adjust the sliders. A degree-3 curve can wiggle at most twice — try to make it turn three times (you cannot). · 调整滑块。三次曲线最多摆动两次——试着让它转三次(你做不到)。
A point where the graph switches from decreasing to increasing is a local . · 图像从递减转为递增的那个点,是一个局部。
A local minimum is a valley — the curve stops falling and starts rising. · 极小值是一个山谷——曲线停止下降,开始上升。
Points of inflection
- Even between turning points, the curve can change the way it bends.
- A point of inflection 拐点 is where the concavity 凹凸性 flips: concave up becomes concave down, or the reverse.
- At an inflection point the rate of change stops speeding up and starts slowing down (or vice versa).
- So it is the steepest or gentlest spot on a rising or falling stretch.
拐点
- 即使在转折点之间,曲线弯曲的方式也可以改变。
- 拐点(point of inflection)是凹凸性(concavity)翻转的地方:凹变凸,或者反过来。
- 在拐点处,变化率停止加快、开始变慢(或反之)。
- 所以它是一段上升或下降曲线上最陡或最缓的位置。
A point of inflection is a point where… · 拐点是这样一个点……
At a point of inflection the curve stops bending one way and starts bending the other — concave up becomes concave down, or vice versa. · 在拐点处,曲线不再朝一个方向弯,而开始朝另一个方向弯——凹变凸,或凸变凹。
Degree caps the wiggles
- A polynomial of degree $n$ has at most $n - 1$ turning points.
- It also has at most $n - 2$ points of inflection.
- A cubic (degree $3$) can turn at most twice; a quartic (degree $4$) at most three times — like the figure above.
- "At most" matters: a curve may have fewer wiggles, but never more than its degree allows.
次数给摆动设了上限
- 一个 $n$ 次多项式最多有 $n - 1$ 个转折点。
- 它也最多有 $n - 2$ 个拐点。
- 三次(次数 $3$)最多转两次;四次(次数 $4$)最多转三次——就像上面的图。
- "最多"很关键:一条曲线可以有更少的摆动,但绝不会超过它的次数所允许的。
At most how many turning points can a polynomial of degree 5 have? · 一个5 次多项式最多能有多少个转折点?
A degree-$n$ polynomial has at most $n - 1$ turning points, so degree $5$ allows at most $4$. · $n$ 次多项式最多有 $n - 1$ 个转折点,所以 5 次最多允许 $4$ 个。
A degree-3 polynomial can have 3 turning points. · 一个 3 次多项式可以有 3 个转折点。
Degree $3$ allows at most $3 - 1 = 2$ turning points. It can have $2$ or none, but never $3$. · 3 次最多允许 $3 - 1 = 2$ 个转折点。它可以有 $2$ 个或没有,但绝不会有 $3$ 个。
Select all · 所有 true statements about polynomial graphs. · 选出关于多项式图像的所有正确说法。
A cubic can have $2$ turning points or none (e.g. $y = x^3$), so "must have 2" is false. The other three are correct. · 三次函数可以有 $2$ 个转折点也可以没有(例如 $y = x^3$),所以"必然有 2 个"是错的。其余三条正确。
"At most" is not "exactly". A cubic can have two turning points, but $y = x^3$ has none — it rises the whole way, with a single point of inflection at the origin. Degree sets the ceiling, not the count.
"最多"不等于"恰好"。三次函数可以有两个转折点,但 $y = x^3$ 一个也没有——它一路上升,只在原点有一个拐点。次数设的是上限,不是实际个数。
Consider $f(x) = x^3 - 3x$.
- Degree $3$, so at most $2$ turning points — and it has exactly $2$: a local maximum near $x = -1$ and a local minimum near $x = 1$.
- Between them, at $x = 0$, the concavity flips: that is its point of inflection.
- Its leading term $x^3$ makes the graph fall on the far left and rise on the far right.
考虑 $f(x) = x^3 - 3x$。
- 次数为 $3$,所以最多 $2$ 个转折点——而它恰好有 $2$ 个:在 $x = -1$ 附近有一个极大值,在 $x = 1$ 附近有一个极小值。
- 在它们之间的 $x = 0$ 处,凹凸性翻转:那就是它的拐点。
- 它的首项 $x^3$ 使图像在最左端下降、在最右端上升。
A polynomial function's degree (the highest power) caps its shape: at most $n - 1$ turning points — the local maxima and minima — and at most $n - 2$ points of inflection, where the concavity flips. The leading term controls what happens far out on each side.
一个多项式函数的次数(最高次幂)给它的形状设了上限:最多 $n - 1$ 个转折点——也就是极大值和极小值——以及最多 $n - 2$ 个拐点(凹凸性翻转处)。首项决定了两端远处的走向。