Key takeaway: End behavior is the direction a graph’s two tails head toward as x runs far left and far right, and for a polynomial the leading term alone settles both directions.
Grade 11 – Algebra 2F-IF.C.7cFunctions and Graphs
Definition of End Behavior
End behavior is the far-away story of a graph: where each tail points once you look at very distant inputs. Small terms of a polynomial fade into the background when \(|x|\) is huge, so the term with the top power decides the whole picture.
Only two clues are needed: the parity of the top power and the sign of its multiplier. An even power sends both tails the same way, like a parabola, while an odd power sends them opposite ways, like a slanted line. A positive multiplier lifts the right tail; a negative one drops it.
Formula and notation
Write end behavior with arrow notation for the two directions:
\[ \text{as } x \to \infty,\ f(x) \to \infty \qquad \text{as } x \to -\infty,\ f(x) \to -\infty \]
For \(f(x) = a_n x^n + \text{lower terms}\), read the sign of \(a_n\) and the parity of \(n\) to fill in both statements.
Key facts
- Even top power with a positive multiplier: both tails point up.
- Even top power with a negative multiplier: both tails point down.
- Odd top power with a positive multiplier: left tail down, right tail up.
- Odd top power with a negative multiplier: left tail up, right tail down.
End Behavior example
Predict how the tails of \(f(x) = -3x^5 + 8x^2 – 40\) and \(g(x) = 2x^4 – 7x^2 + x\) behave far from the origin.
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For \(f\), the leading term is \(-3x^5\): odd degree 5 with a negative coefficient.
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So \(f(x) \to -\infty\) as \(x \to \infty\), and \(f(x) \to \infty\) as \(x \to -\infty\).
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For \(g\), the leading term is \(2x^4\): even degree 4 with a positive coefficient, so \(g(x) \to \infty\) at both ends.
The graph of \(f\) rises on the left and falls on the right; the graph of \(g\) rises on both sides.
Common mistake
Watch out
Cam says \(h(x) = 5 – x^3\) has a positive leading term because the constant 5 is written first.
Why it is wrong
The leading term is the one with the highest power, \(-x^3\), whatever its position in the expression. It has odd degree and a negative coefficient, so the left tail rises and the right tail falls. Rewrite as \(-x^3 + 5\) to see it.
Related terms and lessons
- Polynomial
- Degree of a Polynomial
- Limit at Infinity
- Multiplicity of a Zero
- Zeros of a Function
- Quadratic Function
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