Rational Functions and Vertical Asymptotes · 有理函数与竖直渐近线
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| denominator/dɪˈnɒmɪneɪtə/ | 分母 | fēn mǔ |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
| asymptote/ˈæsɪmptəʊt/ | 渐近线 | jiàn jìn xiàn |
| numerator/ˈnjuːməreɪtə/ | 分子 | fèn zǐ |
| multiplicity/ˌmʌltɪˈplɪsɪti/ | 重数 | chóng shù |
Dividing by almost nothing
- Try $1 \div 0.1 = 10$, then $1 \div 0.01 = 100$, then $1 \div 0.001 = 1000$.
- As the bottom shrinks toward zero, the answer explodes toward infinity.
- A rational function does exactly this near an input that zeroes its denominator.
- The graph rockets up or down along an invisible vertical line.
除以几乎没有的数
- 试试 $1 \div 0.1 = 10$,再 $1 \div 0.01 = 100$,再 $1 \div 0.001 = 1000$。
- 当分母缩向零时,答案朝无穷爆炸。
- 有理函数在让它分母为零的输入附近,做的正是这件事。
- 图像沿着一条看不见的竖直线飞速向上或向下。
Where the denominator hits zero
- A vertical asymptote 竖直渐近线 is a vertical line the graph races along but never crosses.
- It sits at each input that makes the denominator 分母 zero.
- There the function is undefined, and nearby values grow without limit.
- So to find vertical asymptotes, set the bottom equal to zero and solve.
分母触零的地方
- 竖直渐近线(vertical asymptote)是一条图像紧贴飞奔却从不穿过的竖直线。
- 它位于每一个使分母(denominator)为零的输入处。
- 函数在那里无定义,附近的值无限增长。
- 所以要找竖直渐近线,令分母等于零并求解。
A vertical asymptote of a rational function occurs where… · 有理函数的竖直渐近线出现在……
Dividing by a number heading to $0$ makes the value blow up — so a denominator zero (not cancelled) gives a vertical asymptote. · 除以一个趋向 $0$ 的数会让值爆炸——所以未被约掉的分母零点给出一条竖直渐近线。
…unless the factor cancels
- If a value zeroes the numerator 分子 too, the shared factor cancels.
- Then that input is a hole, not an asymptote.
- So a vertical asymptote needs a denominator zero that survives the cancelling.
- Always simplify first, then read the leftover denominator zeros.
……除非因式被约掉
- 如果一个值同时也让分子(numerator)为零,那么公因式会被约掉。
- 这时那个输入是一个空心,而不是渐近线。
- 所以竖直渐近线需要一个在约分后仍然存在的分母零点。
- 永远先化简,再读剩下的分母零点。
Drag the vertical asymptote left and right · 左右拖动竖直渐近线
y = a/(x − b) + c
Change b and the vertical break moves to x = b — the exact input that makes the denominator zero. · 改变 b,竖直的断裂就移到 x = b——正是让分母为零的那个输入。
If a value makes both · 两者 the numerator and denominator zero, there may be a hole instead of a vertical asymptote. · 如果一个值让分子和分母同时为零,那里可能是一个空心,而不是竖直渐近线。
A shared factor cancels, so that input gives a hole (removable). A vertical asymptote needs the denominator zero to survive. · 公因式被约掉,所以那个输入给出一个空心(可去)。竖直渐近线需要分母的零点没被约掉。
Behavior on each side
- An asymptote 渐近线 describes a limiting line; here the limits are infinite.
- Just left of the line, $f(x) \to +\infty$ or $-\infty$; just right, the same choice again.
- Written with limits: $\lim_{x \to 2^-} f(x) = -\infty$ and $\lim_{x \to 2^+} f(x) = +\infty$, for example.
- Checking a value very close on each side tells you which way each branch goes.
两侧的行为
- 渐近线(asymptote)描述一条极限直线;这里的极限是无穷。
- 就在线的左侧,$f(x) \to +\infty$ 或 $-\infty$;右侧,同样是这两种之一。
- 用极限写:例如 $\lim_{x \to 2^-} f(x) = -\infty$,$\lim_{x \to 2^+} f(x) = +\infty$。
- 在两侧各取一个非常靠近的值来检查,就能知道每条分支往哪走。
Near a vertical asymptote, the size of $f(x)$ grows without bound toward . · 在竖直渐近线附近,$f(x)$ 的大小无限增长,趋向。
On each side $f(x) \to +\infty$ or · 或 $-\infty$; the curve races up or down along the asymptote. · 在每一侧 $f(x) \to +\infty$ 或 $-\infty$;曲线沿着渐近线飞速向上或向下。
Multiplicity sets the pattern
- The multiplicity 重数 of the denominator zero decides the two-sided pattern.
- Odd multiplicity → the sign flips across the line: one side $+\infty$, the other $-\infty$.
- Even multiplicity → the sign is the same: both sides go the same way.
- So $\frac{1}{x-2}$ (multiplicity 1) diverges oppositely, while $\frac{1}{(x-2)^2}$ goes up on both sides.
重数决定模式
- 分母零点的重数(multiplicity)决定两侧的模式。
- 奇数重数 → 符号在穿过线时翻转:一侧 $+\infty$,另一侧 $-\infty$。
- 偶数重数 → 符号相同:两侧朝同一方向。
- 所以 $\frac{1}{x-2}$(重数 1)朝相反方向发散,而 $\frac{1}{(x-2)^2}$ 两侧都向上。
If a denominator zero has odd multiplicity, the two sides of the asymptote… · 如果分母的零点是奇数重数,渐近线的两侧……
Odd multiplicity flips the sign as you cross, so the two sides diverge oppositely. Even multiplicity sends both sides the same way. · 奇数重数在穿过时翻转符号,所以两侧朝相反方向发散。偶数重数则让两侧朝相同方向。
Select all · 所有 true statements about vertical asymptotes. · 选出关于竖直渐近线的所有正确说法。
A graph never crosses its vertical asymptote — it only races alongside it. The other three are correct. · 图像从不穿过它的竖直渐近线——只会紧贴着它飞奔。其余三条正确。
A denominator zero is only a vertical asymptote if it does not cancel with the numerator. Check for common factors first — otherwise you will draw an asymptote where there is really just a hole.
一个分母零点只有在它与分子不约分时才是竖直渐近线。先检查公因式——否则你会在其实只是空心的地方画出一条渐近线。
Find the vertical asymptotes of $f(x) = \dfrac{x+1}{(x-3)(x+1)}$.
- Denominator zeros: $x = 3$ and $x = -1$.
- But $x = -1$ also zeroes the numerator, so that factor cancels — a hole, not an asymptote.
- Only $x = 3$ is a vertical asymptote.
求 $f(x) = \dfrac{x+1}{(x-3)(x+1)}$ 的竖直渐近线。
- 分母零点:$x = 3$ 和 $x = -1$。
- 但 $x = -1$ 同时让分子为零,所以那个因式被约掉——是空心,不是渐近线。
- 只有 $x = 3$ 是竖直渐近线。
A vertical asymptote sits at each surviving zero of the denominator — where the graph shoots toward $\pm\infty$. If a factor cancels with the numerator, that input is a hole instead. The multiplicity of the denominator zero decides whether the two sides diverge the same way or oppositely.
竖直渐近线位于分母每一个残留的零点处——图像在那里冲向 $\pm\infty$。如果某个因式与分子约掉,那个输入就变成空心。分母零点的重数决定两侧是朝相同方向还是相反方向发散。