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1.6 · End Behavior
What is end behavior?
What happens to f(x) as x goes way left/right
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1.6 · Limit Notation
Write limit notation for LEFT and RIGHT end behavior.
x goes to -inf and +inf
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1.6 · RIGHT Side Rule
What determines the RIGHT side (x→+∞) of a polynomial's end behavior?
Think: sign of leading coefficient
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1.6 · LEFT Side Rule
How does the LEFT side (x→−∞) relate to the right side?
Think: parity of degree
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1.6 · 4 Cases
State L and R end behavior for all 4 cases: even+pos · even+neg · odd+pos · odd+neg
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1.6 · Example 3d
End behavior of h(x) = 3 − x⁵?
Leading term = −x⁵
Find leading term FIRST
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1.6 · Example 3f
End behavior of m(x) = 2x(x−1)(6−x)?
Expand to find leading term
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1.6 · Example 5b
End behavior of g(x) = (2x−3)/(x+1)?
Rational — not a polynomial
Ratio of leading coefficients
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✓ Limit Notation
Left: lim(x→−∞) f(x) · Right: lim(x→∞) f(x)
Left limit: as x decreases without bound (left tail).
Right limit: as x increases without bound (right tail).
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✓ End Behavior Definition
How f(x) BEHAVES as x → −∞ and x → +∞
End behavior = what happens to the y-values at the far left and far right ends of the graph.
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✓ LEFT Side Rule
Even degree → SAME as right · Odd degree → OPPOSITE of right
Even = same: like a parabola, both ends point the same way.
Odd = opposite: like a cubic, ends go opposite directions.
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✓ RIGHT Side Rule
LC positive → +∞ · LC negative → −∞
Right side follows the sign of the leading coefficient. Positive → right tail up. Negative → right tail down.
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✓ Example 3d — h(x)=3−x⁵
L: +∞ · R: −∞
Leading term −x⁵. Degree 5 (odd), LC=−1 (neg). Right→−∞. Left→opposite=+∞.
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✓ All 4 Cases
Even+pos:↑↑ Even+neg:↓↓ Odd+pos:↓↑ Odd+neg:↑↓
Even=same · Odd=opposite · Pos LC=right goes up · Neg LC=right goes down
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✓ Example 5b — Rational Function
lim(x→±∞) g(x) = 2
Degree num = degree den → limit = ratio of leading coefficients = 2/1 = 2. Horizontal asymptote y=2.
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✓ Example 3f — m(x)=2x(x−1)(6−x)
L: +∞ · R: −∞
Leading term: 2x·x·(−x) = −2x³. Degree 3 (odd), LC=−2 (neg). Right→−∞. Left→+∞.