DWG NO. 2.1 — Lesson 1 of 6

Power Functions and Polynomial End Behavior

Unit 2: Polynomial and Rational Functions · ~20 min

Objective Determine the end behavior of a power function or polynomial from its degree and leading coefficient.

Power functions

A power function has the form f(x) = a\(x^{n}\), where n is a positive whole number. You've already met the simplest ones — f(x) = x² and f(x) = x³ — and their shapes tell you everything about the rest of the family:

Even n (x², x⁴, x⁶, …): both ends of the graph point the same direction, like a parabola. As |x| grows, so does f(x), regardless of the sign of x.
Odd n (x³, x⁵, …): the two ends point opposite directions. Negative inputs and positive inputs send the graph in opposite directions, the way x³ does.
Larger n: the graph flattens near the origin and rises more steeply away from it, compared to a smaller power with the same sign pattern.

End behavior of a polynomial

A full polynomial has more than one term, but far out toward ±∞, the smaller-degree terms become insignificant compared to the term with the highest power — the leading term. That's why a polynomial's end behavior is decided entirely by two things: the degree (the highest exponent) and the sign of the leading coefficient (the number in front of it).

Degree	Leading coeff.	As x → −∞	As x → ∞
Even	Positive	f(x) → ∞	f(x) → ∞
Even	Negative	f(x) → −∞	f(x) → −∞
Odd	Positive	f(x) → −∞	f(x) → ∞
Odd	Negative	f(x) → ∞	f(x) → −∞

This is often called the Leading Coefficient Test. Notice it only needs two pieces of information from a polynomial that might have five or six terms — everything except the leading term is noise once x is far enough from zero.

f(x) = −0.25x⁴ + x² — degree 4 (even), negative leading coefficient: both ends fall

Worked Example 1 · Even degree, positive leading coefficient

ProblemDescribe the end behavior of f(x) = 2x⁴ − 3x² + 1.

1The leading term is 2x⁴: degree 4 (even), leading coefficient +2 (positive).

2Even degree means both ends point the same way; positive leading coefficient means that way is up.

As x → −∞, f(x) → ∞ · As x → ∞, f(x) → ∞

Worked Example 2 · Odd degree, negative leading coefficient

ProblemDescribe the end behavior of f(x) = −3x⁵ + 2x² − 1.

1The leading term is −3x⁵: degree 5 (odd), leading coefficient −3 (negative).

2Odd degree means the ends point opposite ways; negative leading coefficient flips the usual positive-odd pattern.

As x → −∞, f(x) → ∞ · As x → ∞, f(x) → −∞

Worked Example 3 · Reading end behavior from the diagram above

Problemf(x) = −0.25x⁴ + x² has degree 4 (even) and leading coefficient −0.25 (negative).

1By the test, both ends should fall — which matches the diagram: the curve rises and dips in the middle, but both tails point down.

2Notice the middle detail (two humps) doesn't affect the end behavior at all — only the degree and leading coefficient decide that.

As x → −∞, f(x) → −∞ · As x → ∞, f(x) → −∞

Guided practice

1.Describe the end behavior of f(x) = 5x³ − x.
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Degree 3 (odd), leading coefficient +5 (positive). As x → −∞, f(x) → −∞; as x → ∞, f(x) → ∞.
2.Describe the end behavior of f(x) = −2x⁴ + 7x − 1.
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Degree 4 (even), leading coefficient −2 (negative). As x → ±∞, f(x) → −∞ (both ends fall).
3.Describe the end behavior of f(x) = x⁶ − 4x².
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Degree 6 (even), leading coefficient +1 (positive). As x → ±∞, f(x) → ∞ (both ends rise).
4.Describe the end behavior of f(x) = −7x⁵ + 3x³.
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Degree 5 (odd), leading coefficient −7 (negative). As x → −∞, f(x) → ∞; as x → ∞, f(x) → −∞.
5.A polynomial's graph falls to the left and rises to the right. What can you conclude about its degree (odd or even) and the sign of its leading coefficient?
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The two ends point in opposite directions, so the degree must be odd. Falling on the left and rising on the right matches the odd/positive row of the table, so the leading coefficient must be positive.

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