Rational Functions and End Behavior
| English | Chinese | Pinyin |
|---|---|---|
| rational function | 有理函数 | yǒu lǐ hán shù |
| numerator | 分子 | fèn zǐ |
| denominator | 分母 | fēn mǔ |
| horizontal asymptote | 水平渐近线 | shuǐ píng jiàn jìn xiàn |
| slant asymptote | 斜渐近线 | xié jiàn jìn xiàn |
| asymptote | 渐近线 | jiàn jìn xiàn |
| polynomial long division | 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.
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.
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.
Slide the horizontal asymptote up and down
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.
If the denominator has a higher degree than the numerator, the end behavior approaches…
A bigger bottom overpowers the top, so the fraction shrinks toward $0$: the x-axis is a horizontal asymptote.
If numerator and denominator have equal degrees, the horizontal asymptote is…
For $\frac{3x^2+\dots}{x^2+\dots}$ the ends approach $y = 3/1 = 3$ — the ratio of the leading coefficients.
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.
When the numerator degree is exactly one more than the denominator, the graph has a ____ asymptote.
The end behavior follows a tilted line — a slant asymptote — which polynomial long division reveals.
Which end-behavior outcomes are possible for a rational function?
A rational function can have a horizontal asymptote, a slant asymptote, or neither — but never two at the same end.
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.
Polynomial long division can rewrite a rational function to reveal its asymptotic behavior.
Dividing gives "quotient + remainder/denominator"; the quotient is the line or curve the ends follow.
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.
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 $y = 3$.
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.