AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/1/1-6)
Unit 1 · Topic 1.6
1.6 Polynomial Functions and End Behavior
End behavior is what a function does far to the right and far to the left. For a polynomial, the leading term alone decides it, so you only need the degree and the sign of the leading coefficient.
Key terms
- end behavior
- leading term
- limit notation
- degree
Describing end behavior with limits
As x increases without bound (x → ∞), the outputs of a nonconstant polynomial either increase without bound or decrease without bound. The same is true as x decreases without bound (x → −∞). A polynomial never levels off.
Limit notation is how you write this. lim (x→∞) p(x) = ∞ means “as x gets larger and larger, p(x) grows without bound.” lim (x→−∞) p(x) = −∞ means “as x heads to the far left, p(x) decreases without bound.”
Why the leading term wins
For inputs with a large size, the leading term is much bigger than all the other terms put together. Take p(x) = x³ − 100x². At x = 1000, x³ = 1,000,000,000 while 100x² = 100,000,000. The leading term is ten times bigger, and the gap keeps widening as x grows.
So p behaves like its leading term aₙxⁿ at both ends. The degree tells you whether the two ends match, and the sign of the leading coefficient tells you which way.
The four cases
Even degree: both ends point the same way. Odd degree: the ends point opposite ways.
| Degree | Leading coefficient | As x → −∞ | As x → ∞ |
|---|---|---|---|
| Even | Positive | p(x) → ∞ | p(x) → ∞ |
| Even | Negative | p(x) → −∞ | p(x) → −∞ |
| Odd | Positive | p(x) → −∞ | p(x) → ∞ |
| Odd | Negative | p(x) → ∞ | p(x) → −∞ |
Factored form and graphs
When a polynomial is in factored form, find the leading term by multiplying the leading term of each factor. You don't need to expand the whole thing. In (2x − 1)(x + 3)², the leading term is (2x)(x²) = 2x³.
On a graph, the end behavior is the direction of the arrows at the two edges. If a graph's left end goes down and its right end goes up, an odd-degree polynomial with a positive leading coefficient fits.
Sketching with end behavior and zeros
End behavior plus zeros gives you a quick sketch. Start at the far left with the left-end behavior, move right through each real zero (crossing the x-axis at odd multiplicity, touching and turning back at even multiplicity), and finish with the right-end behavior.
One consequence: every odd-degree polynomial has at least one real zero. Its ends point in opposite directions, one below the x-axis and one above, and a polynomial's graph has no breaks, so it has to cross the x-axis somewhere.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
End behavior from standard form
Describe the end behavior of p(x) = −2x⁵ + 3x⁴ − x + 7 using limit notation.
Show the solutionHide the solution
- Step 1: The leading term is −2x⁵: degree 5 (odd) and leading coefficient −2 (negative).
- Step 2: Odd degree means the ends go opposite ways. Negative coefficient means the right end goes down.
- Step 3: Check the left end: for a large negative x, x⁵ is a large negative number, and −2 times it is a large positive number.
Answer: lim (x→∞) p(x) = −∞ and lim (x→−∞) p(x) = ∞.
- Example 2
End behavior from factored form
Describe the end behavior of p(x) = (3 − x)(x + 1)²(2x − 1).
Show the solutionHide the solution
- Step 1: Leading term of each factor: −x from (3 − x), x² from (x + 1)², and 2x from (2x − 1).
- Step 2: Multiply them: (−x)(x²)(2x) = −2x⁴. (Expanding fully gives −2x⁴ + 3x³ + 9x² + x − 3, which agrees.)
- Step 3: Even degree with a negative leading coefficient: both ends go down.
Answer: lim (x→∞) p(x) = −∞ and lim (x→−∞) p(x) = −∞.
- Example 3
Trap: a hidden negative
Describe the end behavior of q(x) = x²(2 − x)³.
Show the solutionHide the solution
- Step 1: The leading term of (2 − x) is −x, not x. So the leading term of (2 − x)³ is (−x)³ = −x³.
- Step 2: Multiply by x²: the leading term of q is −x⁵.
- Step 3: Odd degree, negative leading coefficient: the right end goes down and the left end goes up. A student who reads (2 − x)³ as if it started with +x³ gets both ends backward.
Answer: lim (x→∞) q(x) = −∞ and lim (x→−∞) q(x) = ∞.
Common mistakes
- Taking the first term written as the leading term. In 4 + 2x − x², the leading term is −x², not 4.
- Missing a negative inside a factor like (3 − x) or (1 − 2x) when finding the leading term of a factored polynomial.
- Saying a polynomial approaches a horizontal asymptote. Nonconstant polynomials always grow or fall without bound at both ends.
On the exam
- Expect to match a polynomial to a graph or description using end behavior plus intercepts. Write end behavior in limit notation when asked.
- On free-response questions, end behavior often appears as one part of a function-concepts question: state both limits clearly.
Connected topics
Videos
Check yourself
4 questions on 1.6 Polynomial Functions and End Behavior. Pick an answer to see if you got it, and why.
Let p(x) = −2x⁵ + 3x⁴ − x + 7. Which of the following describes the end behavior of p?
Let p(x) = −(x − 1)²(x + 2)³(2x − 3). Which of the following describes the end behavior of p?
Which of the following polynomial functions satisfies lim (x→−∞) p(x) = −∞ and lim (x→∞) p(x) = −∞?
A polynomial function p has degree n and leading coefficient a. Both lim (x→−∞) p(x) and lim (x→∞) p(x) equal ∞. Which of the following could be true?
0 of 4 answered