End Behavior of the Graph of a Polynomial
End Behavior – what happens when 𝒙 gets very large, approaching −∞ and approaching +∞?
𝒇(𝒙) = 𝒂𝒏 𝒙𝒏 + 𝒂𝒏−𝟏 𝒙𝒏−𝟏 + ⋯ + 𝒂𝟏 𝒙 + 𝒂𝟎
 This is a POLYNOMIAL. Its degree is 𝑛 and its leading coefficient is 𝑎𝑛 .
 The degree and the leading coefficient tell you about the End Behavior of its graph:
If the degree is ODD
and the leading
and the leading
coefficient is positive
coefficient is negative
If the degree is EVEN
and the leading
and the leading
coefficient is positive
coefficient is negative
It behaves like 𝑦 = 𝑥 3
It behaves like 𝑦 = 𝑥 2
lim 𝑓(𝑥) = −∞
𝑥→−∞
lim 𝑓(𝑥) = ∞
𝑥→+∞
It’s like 𝑦 = −𝑥 3
lim 𝑓(𝑥) = ∞
𝑥→−∞
lim 𝑓(𝑥) = −∞
𝑥→+∞
lim 𝑓(𝑥) = ∞
𝑥→−∞
lim 𝑓(𝑥) = ∞
𝑥→+∞
It’s like 𝑦 = −𝑥 2
lim 𝑓(𝑥) = −∞
𝑥→−∞
lim 𝑓(𝑥) = −∞
𝑥→+∞
NOTE: This only applies to the End Behavior. What happens in the middle depends on the rest of the polynomial.
M1111 QuickNotes/PolynomialEndBehavior.docx 6/23/2016 5:41 PM D.R.S.