Mathfolis

Polynomial Functions and End Behavior

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

The end behavior of a polynomial is governed entirely by its leading term. The four cases line up by sign of the leading coefficient and parity of the degree: same infinity on both ends when the degree is even, opposite ends when the degree is odd. AP Precalc tests this both forward (state the end behavior) and backward (match a described graph to a possible degree and leading sign).

The four cases (with leading coefficient ana_n and degree n)

an>0a_n > 0, n even
x→±∞  ⇒  f→+∞x \to \pm\infty \;\Rightarrow\; f \to +\infty
an>0a_n > 0, n odd
x→−∞  ⇒  f→−∞;  x→+∞  ⇒  f→+∞x \to -\infty \;\Rightarrow\; f \to -\infty;\; x \to +\infty \;\Rightarrow\; f \to +\infty
an<0a_n < 0, n even
x→±∞  ⇒  f→−∞x \to \pm\infty \;\Rightarrow\; f \to -\infty
an<0a_n < 0, n odd
x→−∞  ⇒  f→+∞;  x→+∞  ⇒  f→−∞x \to -\infty \;\Rightarrow\; f \to +\infty;\; x \to +\infty \;\Rightarrow\; f \to -\infty
AP Tip: Only the leading term matters. Constants and middle coefficients fall away as |x| → ∞ — but watch (a − x) factors, which contribute a sign flip to the leading term.
Type 1

End behavior from standard form

Identify the leading term, then read off the parity of the degree and sign of the coefficient.

Example 1
State the end behavior of f(x)=−3x4+5x2−1f(x) = -3x^4 + 5x^2 - 1. (A) f→+∞f \to +\infty on both ends (B) f→−∞f \to -\infty on both ends (C) f→+∞f \to +\infty as x→−∞x \to -\infty, −∞-\infty as x→+∞x \to +\infty (D) f→−∞f \to -\infty as x→−∞x \to -\infty, +∞+\infty as x→+∞x \to +\infty

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 2

End behavior from factored form

Multiply the leading factors of each parenthesized term to recover the leading term, then apply the four-case rule.

Example 2
State the end behavior of f(x)=2(x−1)(x+3)(x−5)f(x) = 2(x - 1)(x + 3)(x - 5). (A) Both ends to +∞+\infty (B) Both ends to −∞-\infty (C) Left to −∞-\infty, right to +∞+\infty (D) Left to +∞+\infty, right to −∞-\infty

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 3

Match degree parity and leading sign to a graph description

Given a description of the graph's end behavior, identify the parity of the degree (do ends match?) and the sign of the leading coefficient (which infinity?).

Example 3
A polynomial graph has f→−∞f \to -\infty as x→−∞x \to -\infty and f→+∞f \to +\infty as x→+∞x \to +\infty. What can you say about the degree parity and leading sign? (A) Even degree, positive leading coefficient (B) Even degree, negative leading coefficient (C) Odd degree, positive leading coefficient (D) Odd degree, negative leading coefficient

Practice more of this type— AI-generated · always-new problems

Generate Problems →