Family Overview

In Plain Language

A rational function is a quotient of polynomials, $r(x)=\dfrac{p(x)}{q(x)}$ (with $q\not\equiv 0$). For large $|x|$, each polynomial acts like its leading term, so end behavior comes from the quotient of the leading terms.

Compare degrees:

• Denominator degree larger → denominator dominates → horizontal asymptote $y=0$.
• Equal degrees → neither dominates → horizontal asymptote $y=$ ratio of leading coefficients.
• Numerator degree larger → numerator dominates → end behavior like that nonconstant quotient polynomial; if the degree difference is exactly 1, there is a slant (oblique) asymptote.

A horizontal asymptote $y=b$ means outputs get (and stay) arbitrarily close to $b$ as $x\to\infty$ or $x\to-\infty$: $\lim_{x\to\infty}r(x)=b$ (and/or $x\to-\infty$).

Why It Matters

AP Precalculus moves from polynomial ends (1.6) to rational ends by asking “who dominates?” Graphing and modeling both need the correct horizontal/slant story, not just hole/asymptote vertical ideas (later topics).

How Families Can Help

• Ask: “What’s the degree on top? On bottom?”
• Practice three cards: bottom wins → $y=0$; tie → ratio; top wins → no HA (maybe slant).
• Have them say the limit sentence when $y=b$ is an asymptote.

What Not to Do

• Do not use only the constant terms to find horizontal asymptotes.
• Do not assume every rational has $y=0$ as HA.
• Do not confuse vertical asymptotes (zeros of denominator) with end behavior.
• Do not skip the slant case when deg(num)=deg(den)+1.