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College Algebra

Polynomial End Behavior

Lesson

End behavior describes what a function does at the far left (as x→−∞x \to -\infty) and the far right (as x→∞x \to \infty) of its graph. For a polynomial, only one term matters at the extremes: the leading term — the term with the highest power of xx. Everything else is rounding error compared to it once xx gets large.

Two things about the leading term decide everything:

  • The degree (the exponent on xx) — even or odd?
  • The leading coefficient — positive or negative?

The four cases:

  • Even degree, positive coefficient (like x2x^2): both ends go up.
  • Even degree, negative coefficient (like −x2-x^2): both ends go down.
  • Odd degree, positive coefficient (like x3x^3): down on the left, up on the right.
  • Odd degree, negative coefficient (like −x3-x^3): up on the left, down on the right.

Worked example 1

f(x)=−2x4+3x2+1f(x) = -2x^4 + 3x^2 + 1

Leading term: −2x4-2x^4. Degree 4 (even), coefficient −2 (negative). Even/negative → both ends down.

End behavior: down, down\text{End behavior: down, down}

Worked example 2

f(x)=5x−2x3+1f(x) = 5x - 2x^3 + 1

The leading term is not always written first. Find the highest power of xx: it’s −2x3-2x^3. Degree 3 (odd), coefficient −2 (negative). Odd/negative → up on the left, down on the right.

End behavior: up, down\text{End behavior: up, down}

How to type your answer

Type the left end, a comma, then the right end. Use the words up or down. Examples: up,up, down,up, up,down.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x)=x2f(x) = x^2

Problem 2

f(x)=x3f(x) = x^3

Problem 3

f(x)=−x2f(x) = -x^2

Problem 4

f(x)=−x3f(x) = -x^3

Practice

Standard problems matching the lesson.

Problem 5

f(x)=x4f(x) = x^4

Problem 6

f(x)=x5f(x) = x^5

Problem 7

f(x)=−x4f(x) = -x^4

Problem 8

f(x)=−x5f(x) = -x^5

Problem 9

f(x)=3x2+5x−1f(x) = 3x^2 + 5x - 1

Problem 10

f(x)=−2x2+7f(x) = -2x^2 + 7

Problem 11

f(x)=2x3−4xf(x) = 2x^3 - 4x

Problem 12

f(x)=−x3+3x2f(x) = -x^3 + 3x^2

Problem 13

f(x)=4x4−x2+3f(x) = 4x^4 - x^2 + 3

Problem 14

f(x)=−5x4+x2f(x) = -5x^4 + x^2

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

f(x)=5x−2x3+1f(x) = 5x - 2x^3 + 1

Problem 16

f(x)=−x2+x5−3f(x) = -x^2 + x^5 - 3

Problem 17

f(x)=2−3x6f(x) = 2 - 3x^6

Problem 18

f(x)=x+x7f(x) = x + x^7

Problem 19

f(x)=−4+5x2−2x3+xf(x) = -4 + 5x^2 - 2x^3 + x

Problem 20

f(x)=(x−1)(x+2)(x−3)f(x) = (x - 1)(x + 2)(x - 3)

Problem 21

f(x)=−(x+1)2(x−2)f(x) = -(x + 1)^2(x - 2)

Problem 22

f(x)=(x−1)(x+1)(x2+4)f(x) = (x - 1)(x + 1)(x^2 + 4)

Practice

Standard problems matching the lesson.

Problem 23

End behavior of y = x³ − 5x² + 7x − 2 (left,right) using down/up.

Problem 24

End behavior of y = -2x⁴ (left,right) using down/up.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

End behavior of y = 3x⁵ − 7x² + 1 (left,right) using down/up.

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