1.6 End Behavior — AP Precalculus | The Complete Equation

AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS

1.6 — Polynomial Functions and End Behavior

Notes — What Happens at the Far Left and Far Right

💡 Learning Objectives (1.6.A.1 – 1.6.A.3)

By the end of this lesson you will be able to:

Describe what a polynomial does as the input grows without bound (x → ∞) and as it decreases without bound (x → −∞)
Use the standard limit notation: lim p(x) = ∞ or lim p(x) = −∞ as x → ∞ or x → −∞
Predict end behavior from the degree (even or odd) and sign of the leading coefficient
Explain why the leading term dominates all lower-degree terms for large |x|

1. What ‘End Behavior’ Means

End behavior describes how the output of a function behaves at the two far ends of its domain — that is, what happens to f(x) as x grows extremely large in either the positive or the negative direction.

For polynomials, only two outcomes are possible at each end: the function shoots up without bound (toward +∞), or it plunges down without bound (toward −∞). It will never level off, oscillate, or reverse direction once x is large enough.

2. Limit Notation

💡 How we write end behavior

To say that p(x) grows without bound as x grows without bound, we write:

lim p(x) = ∞ as x → ∞.

To say that p(x) plunges to −∞ as x grows toward +∞:

lim p(x) = −∞ as x → ∞.

Similar notation describes the left end as x → −∞.

3. The Leading Term Dominates

Why does end behavior depend only on the leading term? Look at p(x) = x³ − 4x + 2. When x = 100, the term x³ contributes 1,000,000, while −4x + 2 contributes only −398. The cubic term dwarfs the rest. As x grows even larger, the gap widens — the lower-degree terms become irrelevant in comparison to the leading term.

For large |x|, the polynomial p(x) = x³ − 4x + 2 (blue) is virtually indistinguishable from its leading term x³ (red dashed). Lower-degree terms only matter near the origin.

💡 Leading term rule

The end behavior of a polynomial is identical to the end behavior of its leading term aₙxⁿ.

So to determine end behavior, you only need two pieces of information:

Is the degree n even or odd?
Is the leading coefficient aₙ positive or negative?

4. The Four Cases

The four possible end-behavior patterns for a polynomial, depending on parity of degree and sign of the leading coefficient. The behavior near the origin can be complicated, but the two ends are always one of these four shapes.

Degree	Leading coeff	x → ∞	x → −∞
even (e.g. 2, 4, 6, …)	positive	+∞	+∞
even	negative	−∞	−∞
odd (e.g. 1, 3, 5, …)	positive	+∞	−∞
odd	negative	−∞	+∞

📘 Example — Predicting end behavior

For p(x) = −3x⁴ + 2x³ − 7:

Leading term is −3x⁴. Degree 4 is even, leading coefficient −3 is negative.

→ Both ends head to −∞.

In limit notation: lim p(x) = −∞ as x → ∞, and lim p(x) = −∞ as x → −∞.

📘 Example — Another

For q(x) = 2x⁵ − x² + 8:

Leading term 2x⁵. Degree 5 is odd, leading coefficient +2 is positive.

→ Right end goes to +∞; left end goes to −∞.

In limit notation: lim q(x) = ∞ as x → ∞ and lim q(x) = −∞ as x → −∞.

5. Strategy: From a Graph to End Behavior

If you are given a graph and asked to describe end behavior, follow your eye from the centre out to each side:

Far right (x → ∞): what is the curve doing — climbing up or diving down?
Far left (x → −∞): is the curve climbing or diving as you move left?
If both ends do the same thing (both up or both down), the polynomial has even degree.
If the ends do opposite things, the polynomial has odd degree.
Up at the far right indicates a positive leading coefficient (regardless of parity).

6. Strategy: From End Behavior to a Possible Polynomial

📘 Example — Designing a polynomial with given end behavior

Find a polynomial whose graph rises to +∞ on the right and rises to +∞ on the left.

Both ends up ⇒ even degree, positive leading coefficient.

Lowest possible: a degree-2 polynomial like p(x) = x². Or degree 4: p(x) = x⁴ − 3x² + 2. Many possibilities.

📘 Example — Another

Find a polynomial whose left end goes to +∞ and right end goes to −∞.

Opposite ends ⇒ odd degree. Right end down ⇒ negative leading coefficient.

Lowest possible: p(x) = −x. Or p(x) = −x³ + 2x. Or p(x) = −2x⁵ + x − 1.

7. End Behavior and Degree Parity at a Glance

💡 Pattern to memorize

Even degree, positive leading coeff → ‘both ends UP’ (think x² or x⁴).
Even degree, negative leading coeff → ‘both ends DOWN’ (think −x² or −x⁴).
Odd degree, positive leading coeff → ‘DOWN on the left, UP on the right’ (think x³ or x⁵).
Odd degree, negative leading coeff → ‘UP on the left, DOWN on the right’ (think −x³ or −x⁵).

⚠️ End behavior tells you only what happens at the ends

Don't confuse end behavior with what the polynomial does in the middle. A polynomial whose ends both go to +∞ can still dive deeply into negative territory in the middle (think x⁴ − 10x² = x²(x² − 10) which is negative on (−√10, 0) and (0, √10) — but climbs to +∞ on both ends).

8. Summary

End behavior describes what happens to f(x) as x → ±∞
Polynomial end behavior is fully determined by the leading term
Even degree ⇒ both ends behave the same (both up or both down); odd degree ⇒ ends behave oppositely
Sign of the leading coefficient flips ‘up’ to ‘down’ at the right end
Limit notation: lim p(x) = ∞ or = −∞ as x → ∞ or x → −∞