Rational Functions and End Behavior · 有理函数与末端行为
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rational function/ˈræʃənl ˈfʌŋkʃn/ | 有理函数 | yǒu lǐ hán shù |
| numerator/ˈnjuːməreɪtə/ | 分子 | fèn zǐ |
| denominator/dɪˈnɒmɪneɪtə/ | 分母 | fēn mǔ |
| horizontal asymptote/ˌhɒrɪˈzɒntl ˈæsɪmptəʊt/ | 水平渐近线 | shuǐ píng jiàn jìn xiàn |
| slant asymptote/slænt ˈæsɪmptəʊt/ | 斜渐近线 | xié jiàn jìn xiàn |
| asymptote/ˈæsɪmptəʊt/ | 渐近线 | jiàn jìn xiàn |
| polynomial long division/ˌpɒlɪˈnəʊmɪəl lɒŋ dɪˈvɪʒn/ | 多项式长除法 | duō xiàng shì zhǎng chú fǎ |
A tug-of-war between top and bottom
- A fraction like $\frac{2x^2}{x^2+1}$ has a polynomial pulling from the top and one from the bottom.
- Far from the origin, whichever grows faster decides where the value settles.
- The result is often a flat line the graph hugs but never quite reaches.
- Reading that line is how you describe a rational function's ends.
上下两端的拔河
- 像 $\frac{2x^2}{x^2+1}$ 这样的分数,上面有一个多项式在拉,下面也有一个在拉。
- 在远离原点处,谁增长得更快,就决定了值最终停在哪里。
- 结果常常是一条图像贴近却始终触不到的水平线。
- 读出那条线,就是描述有理函数末端的方法。
What a rational function is
- A rational function 有理函数 is one polynomial divided by another: $\frac{\text{numerator}}{\text{denominator}}$.
- The numerator 分子 is the top; the denominator 分母 is the bottom.
- Its end behavior is a contest between the two degrees.
- Compare the highest powers of top and bottom — that alone sets the ends.
什么是有理函数
- 有理函数(rational function)是一个多项式除以另一个:分子 / 分母。
- 分子(numerator)在上面;分母(denominator)在下面。
- 它的末端行为是两个次数之间的较量。
- 比较上下的最高次幂——单凭这一点就定下了两端。
Compare the degrees
- Bottom bigger: the fraction shrinks toward $0$, so $y = 0$ is a horizontal asymptote 水平渐近线.
- Equal degrees: the ends approach the ratio of the leading coefficients.
- Top bigger by one: the ends follow a slant asymptote 斜渐近线 (a tilted line).
- Top bigger by more: no asymptote — it grows like a polynomial.
比较次数
- 分母更大:分数缩向 $0$,所以 $y = 0$ 是一条水平渐近线(horizontal asymptote)。
- **次数相等:**两端趋近于首项系数之比。
- 分子高一次:两端沿着一条斜渐近线(slant asymptote)(一条倾斜的线)。
- **分子高更多:**没有渐近线——它像多项式一样增长。
Slide the horizontal asymptote up and down · 上下滑动水平渐近线
y = a/(x − b) + c
Change c and the whole curve levels off at a new height y = c far out — that horizontal line is the end behavior. · 改变 c,整条曲线在远处就会稳定在新的高度 y = c——那条水平线就是末端行为。
If the denominator has a higher degree than the numerator, the end behavior approaches… · 如果分母的次数高于分子,末端行为趋近于……
A bigger bottom overpowers the top, so the fraction shrinks toward $0$: the x-axis is a horizontal asymptote. · 更大的分母压过分子,所以分数缩向 $0$:x 轴是一条水平渐近线。
If numerator and denominator have equal · 相等 degrees, the horizontal asymptote is… · 如果分子与分母的次数相等,水平渐近线是……
For · 支持 $\frac{3x^2+\dots}{x^2+\dots}$ the ends approach $y = 3/1 = 3$ — the ratio of the leading coefficients. · 对 $\frac{3x^2+\dots}{x^2+\dots}$,两端趋近 $y = 3/1 = 3$——即首项系数之比。
The horizontal or slant line
- An asymptote 渐近线 is a line the graph approaches but never crosses far out.
- Same degree → horizontal asymptote at a non-zero height; bottom wins → horizontal asymptote at $y = 0$.
- Degree of top one more than bottom → a slant asymptote instead of a horizontal one.
- Either way, the ends of the graph settle onto that line.
那条水平线或斜线
- 渐近线(asymptote)是图像在远处靠近却从不越过的一条线。
- 次数相等 → 在非零高度处有水平渐近线;分母获胜 → 水平渐近线在 $y = 0$。
- 分子次数比分母高一次 → 变成斜渐近线,而不是水平渐近线。
- 无论哪种,图像的两端都会停靠在那条线上。
When the numerator degree is exactly one more than the denominator, the graph has a ____ asymptote. · 当分子次数恰好比分母高一次时,图像有一条____渐近线。
The end behavior follows a tilted line — a slant asymptote — which polynomial long division reveals. · 末端行为沿着一条倾斜的直线——斜渐近线——多项式长除法能把它揭示出来。
Which end-behavior outcomes are possible for a rational function? · 有理函数可能出现哪些末端行为?
A rational function can have a horizontal asymptote, a slant asymptote, or neither — but never two at the same end. · 有理函数可以有水平渐近线、斜渐近线或都没有——但同一端绝不会有两条。
Long division makes it obvious
- Polynomial long division 多项式长除法 rewrites $\frac{\text{top}}{\text{bottom}}$ as quotient $+$ remainder/bottom.
- Far out, the remainder term vanishes, leaving just the quotient.
- If the quotient is a number, that is the horizontal asymptote; if it is a line, that is the slant asymptote.
- Limits say the same thing: $\lim_{x\to\infty} f(x)$ equals that quotient.
长除法让它一目了然
- 多项式长除法(polynomial long division)把 上 / 下 改写成 商 $+$ 余数 / 下。
- 在远处,余数那一项消失,只剩下商。
- 如果商是一个数,那就是水平渐近线;如果商是一条线,那就是斜渐近线。
- 极限也说同样的话:$\lim_{x\to\infty} f(x)$ 就等于那个商。
Polynomial long division can rewrite a rational function to reveal its asymptotic behavior. · 多项式长除法可以把有理函数改写,从而揭示它的渐近行为。
Dividing gives "quotient + remainder/denominator"; the quotient is the line or curve the ends follow. · 相除得到"商 + 余数/分母";那个商就是两端所沿的直线或曲线。
Only the leading terms of the top and bottom control end behavior — the smaller terms fade away far from the origin. So compare degrees first; do not be distracted by the constants or middle terms.
只有上下的首项控制末端行为——较小的项在远离原点处逐渐消失。所以先比较次数;不要被常数项或中间的项分散注意力。
Find the end behavior of $f(x) = \dfrac{3x^2 - 5}{x^2 + 4}$.
- Top and bottom both have degree $2$ — an equal-degree tie.
- The horizontal asymptote is the ratio of leading coefficients: $y = \frac{3}{1} = 3$.
- So $\lim_{x \to \pm\infty} f(x) = 3$: the graph flattens toward the line $y = 3$.
求 $f(x) = \dfrac{3x^2 - 5}{x^2 + 4}$ 的末端行为。
- 上下都是 $2$ 次——次数相等的平局。
- 水平渐近线是首项系数之比:$y = \frac{3}{1} = 3$。
- 所以 $\lim_{x \to \pm\infty} f(x) = 3$:图像趋平,靠向 $y = 3$ 这条线。
For a rational function, compare the degrees of numerator and denominator: bottom bigger → horizontal asymptote $y = 0$; equal → horizontal asymptote at the ratio of leading coefficients; top bigger by one → slant asymptote. Polynomial long division reveals the line the ends follow.
对有理函数,比较分子和分母的次数:分母更大 → 水平渐近线 $y = 0$;相等 → 水平渐近线在首项系数之比处;分子高一次 → 斜渐近线。多项式长除法能揭示两端所沿的那条线。