End behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.
In Spanish
comportamiento en los extremos
Also called
end behaviour, long-run behavior, behavior as x goes to infinity
Degree even or odd decides whether the ends match; the sign of the leading coefficient decides which way the right end points.
Formal definition and notation
The end behavior of a functionf is what happens to f(x) as x → ∞ and as x → −∞: it may grow without bound (f(x) → ∞), drop without bound (f(x) → −∞), or level off toward a number. For a polynomialf(x) = anxn + ⋯ + a0 with an ≠ 0, the leading term anxn is much larger than the others when |x| is large, so f has the same end behavior as anxn. Even n: both ends point the same way, up if an > 0 and down if an < 0. Odd n: the ends point opposite ways, falling left and rising right if an > 0, and the reverse if an < 0.
Symbol
Read it as
Meaning
x → ∞
x approaches infinity
moving to the far right of the graph
x → −∞
x approaches negative infinity
moving to the far left of the graph
f(x) → ∞
f of x approaches infinity
the graph rises without bound (the end points up)
Key facts
Only the leading term matters for the ends. The other terms change the wiggles in the middle, not where the graph goes far away.
Find the leading term first: in f(x) = 3 + 2x − x4, it is −x4, even and negative, so both ends go down.
In factored form, multiply the leading parts of each factor: (x − 1)2(3 − x) starts with x2 · (−x) = −x3.
Even-degree polynomials have both ends on the same side; odd-degree polynomials have ends on opposite sides, so they always cross the x-axis at least once.
A constant function, such as f(x) = 5, stays at 5 at both ends.
Other functions can level off: f(x) = 2x approaches 0 as x → −∞ and rises as x → ∞.
End behavior rule for polynomials
End behavior rule for polynomials
for large |x|, f(x) behaves like its leading term anxn
n even, an > 0
as x → ±∞, f(x) → ∞: up on both ends, like x2
n even, an < 0
as x → ±∞, f(x) → −∞: down on both ends, like −x2
n odd, an > 0
down on the left, up on the right, like x3
n odd, an < 0
up on the left, down on the right, like −x3
Write the polynomial in standard form, or multiply the leading terms of its factors, before you read n and an.
The leading coefficient is one number, an, the coefficient of the highest power. End behavior is the result of using it together with the degree: in −3x4, the leading coefficient is −3 and the ends both point down.
The degree is the highest exponent, such as 5 in x5 − x. Its parity (even or odd) is half of the end behavior rule; the sign of the leading coefficient is the other half.
End Behaviorvs.Turning points
Turning points are where the graph changes from rising to falling in the middle. They depend on all the terms. End behavior ignores the middle and looks only at the far left and far right.
Quick check: 3 questions
Question 1 of 3
What is the end behavior of f(x) = −x6 + 4x2?
Show answer and explanation
Answer: B. The leading term −x6 has even degree and a negative coefficient, so both ends fall. Up on both ends ignores the negative sign, and the mixed choices treat 6 as odd.
Question 2 of 3
Which function falls to the left and rises to the right?
Show answer and explanation
Answer: C.5x3 is odd with a positive coefficient: down on the left, up on the right. −2x3 + x does the opposite, x4 − 1 rises on both ends, and −x2 + 9 falls on both ends.
Question 3 of 3
For f(x) = 2x(x − 3)(4 − x), what happens to f(x) as x → ∞?
Show answer and explanation
Answer: A. The leading parts multiply to 2x · x · (−x) = −2x3, odd and negative, so the right end falls. ∞ misses the sign in (4 − x); polynomials of degree 1 or more never level off at 0 or 2.
End Behavior: frequently asked questions
How do you find the end behavior of a polynomial?
Find the leading term, the one with the highest power of x. Check two things: is the degree even or odd, and is the coefficient positive or negative? Even and positive: up, up. Even and negative: down, down. Odd and positive: down on the left, up on the right. Odd and negative: up on the left, down on the right.
How do you write end behavior with arrows?
Describe each end separately. For f(x) = x3, write: as x → −∞, f(x) → −∞, and as x → ∞, f(x) → ∞. Read the first arrow as x approaches and the second as f of x approaches. Some books write the same thing in words: falls to the left, rises to the right.
Why don't the other terms affect end behavior?
Because the highest power grows much faster than the rest. For f(x) = x3 − 100x at x = 1000, x3 is 1,000,000,000 while 100x is only 100,000, so the cube term swamps it. Far enough out, the graph looks just like x3, even though the −100x term adds bumps near the middle.
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