Take f(x) = 2x3 − 5x2 + 100. Near the origin the +100 is by far the biggest piece — at x = 1 the whole function is 97, and the leading term contributes only 2 of that. It looks like the 100 is in charge. Watch what happens when x grows.
| x = | 1 | 10 | 100 | 1,000 | 10,000 |
|---|---|---|---|---|---|
| leading term 2x3 | 2 | 2,000 | 2,000,000 | 2×109 | 2×1012 |
| everything else | 95 | −400 | −49,900 | −4,999,900 | −5×108 |
| whole f(x) | 97 | 1,600 | 1,950,100 | 1,995,000,100 | 1,999,500,000,100 |
| f(x) ÷ 2x3 | 48.5 | 0.8 | 0.975 | 0.9975 | 0.99975 |
The bottom row is the whole story. Near x = 1 the ratio is wildly off. By x = 100 the function is within 3% of its leading term; by x = 10,000 it is within 0.03%. The other terms never stop existing — −5×108 is not a small number — they just stop being a meaningful share of the total.
The algebra says exactly why. Factor x3 out of every term:
As |x| grows without bound, 5/x and 100/x3 both shrink toward 0, so the bracket closes in on the single number 2. The function becomes 2 × x3, which is the leading term. Every lower-degree term ends up divided by a power of x, and that is why every one of them is eventually negligible. This works for any polynomial — factor out xn and all that survives is the leading coefficient.
The right arm is easy: for large positive x, xn is positive for every n, so the sign of a alone decides. The left arm is where parity enters, because there you are raising a negative number to the n-th power. Count the minus signs by expanding:
Two negatives multiply to a positive. An even exponent means the minus signs pair off completely and nothing survives — so a negative input gives the same answer as its positive twin, and both ends of the graph do the same thing. An odd exponent leaves exactly one minus stranded, so a negative input flips the sign, and the two ends do opposite things. Concretely:
| n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 2n | 2 | 4 | 8 | 16 | 32 | 64 |
| (−2)n | −2 | 4 | −8 | 16 | −32 | 64 |
| ends agree? | no | yes | no | yes | no | yes |
Now multiply by a. A positive a leaves both arms alone; a negative a flips both arms at once, because multiplying by a negative reverses the sign of every output. That is the whole rule, and it is why there are exactly four possibilities.
Read them as a pair of yes/no answers rather than four separate facts to memorize. Even degree → the arms match. Odd degree → the arms disagree. Then negative a turns the whole picture upside down. If you can picture y = x2 and y = x3, you already know all four — every other polynomial is one of those two shapes at the ends, however busy its middle.
Two notations are in common use and both are expected to be readable:
The symbol ∞ is not a number and f(x) never equals it. The arrow is shorthand for “grows past any bound you care to name.” Writing f(∞) = ∞ is not meaningful, and neither is subtracting one ∞ from another.
It describes only the far left and far right. Everything between is unconstrained, so resist reading more into it than it says:
Not the number of x-intercepts. A degree-n polynomial has anywhere from 0 to n real roots. Both x2 + 1 and x2 − 4 have two arms pointing up; the first never touches the x-axis and the second crosses it twice.
Not the number of turning points — only a ceiling on it. A polynomial of degree n has at most n − 1 turning points, and it may have fewer. Degree 4 permits three turns; y = x4 uses only one.
Not the sign in the middle. Case 2 in the stepper has both arms diving to −∞ yet sits at f(0) = 7, comfortably above the axis. Both facts are true at once, and together they force the graph to cross the x-axis at least twice — which is how end behavior earns its keep: combined with one or two known points, it tells you where crossings must exist.
And it does not describe non-polynomials. Rational functions — the second half of this chapter — can flatten toward a horizontal asymptote instead of running off to ±∞. For (3x2 + 1)/(x2 − 4) the same “divide by the highest power” trick gives a finite limit of 3, not an arm. Everything on this page assumes a genuine polynomial: whole-number exponents only, no variable in a denominator, nothing under a radical.
Domain: all real numbers, with no excluded values. A polynomial is built from nothing but multiplication and addition, so there is no denominator that could be zero and no even root that could go negative. Every real x is legal, which is exactly why both arms run forever and why the graph is a single unbroken curve with no holes, jumps, or vertical asymptotes. Excluded values become a live issue in the very next sections, where a denominator appears — and there, values like x = 2 for 1/(x − 2) must be thrown out before anything else happens.
No equation is being solved here, so no extraneous solutions can appear. Reading end behavior is evaluation and comparison, not solving — nothing gets squared and no denominator gets cleared, and those are the two operations that manufacture false roots. The check still matters the moment you go looking for the intercepts that fill in the middle of the sketch. If you find them by factoring, never divide both sides by a variable expression: dividing x3 = 4x by x silently deletes the perfectly good root x = 0, leaving only ±2. Move everything to one side instead — x3 − 4x = x(x − 2)(x + 2) = 0 — and all three roots survive. And if the polynomial arrived from a word problem, substitute each root back into the original situation: a root meaning a negative length, a negative time, or a fractional person is discarded on meaning even when the algebra is flawless.
The one requirement on a is a ≠ 0. The degree of a polynomial is defined as the exponent whose coefficient is nonzero. Drag a to 0 above and the xn term vanishes outright — what is left is not a degree-n polynomial with flat arms, it is a lower-degree polynomial whose own leading term now runs the show. Stay alert when a coefficient carries a parameter: (m − 5)x4 + 2x3 − 1 has two arms up for m > 5 and two arms down for m < 5, but at exactly m = 5 it collapses to the cubic 2x3 − 1, whose arms point in opposite directions.