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Polynomial Graphs and End Behavior

A free Algebra II lesson from the “Polynomial Graphs” unit, with a worked example and practice problems including step-by-step solutions.

A polynomial's degree and leading coefficient predict what the graph does far to the left and far to the right. Even degree graphs have matching end directions; odd degree graphs have opposite end directions.

What you'll learn

Why it matters: Population-growth projections, stress curves, and tax-bracket cumulative graphs all eventually behave like their leading term. End behavior is the long-range forecast — the answer to 'what happens far from the visible window?'

Worked example

Problem. Describe the end behavior of y = -2x^4 + 3x - 1.

  1. The degree is even.
  2. The leading coefficient is negative.
  3. Even degree with negative leading coefficient means both ends go down.

Answer: both ends down

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. The graph of y = x^4 has ends that...

Choices: Both rise · Both fall · Left rises and right falls · Left falls and right rises

Show solution
  1. Even degree and positive leading coefficient.

Answer: Both rise

2. The graph of y = -x^3 has ends that...

Choices: Left rises and right falls · Both rise · Both fall · Left falls and right rises

Show solution
  1. Odd degree and negative leading coefficient.

Answer: Left rises and right falls

3. For y = 5x^6 - x + 1, the end behavior is...

Choices: Both rise · Both fall · Left rises and right falls · Left falls and right rises

Show solution
  1. Even degree with positive leading coefficient.

Answer: Both rise

4. For y = -3x^5 + 2x^2, the right end...

Choices: Falls · Rises · Stays flat · Crosses twice

Show solution
  1. Odd degree with negative leading coefficient falls right.

Answer: Falls

5. What is the degree of 4x^5 - 2x^2 + 1?

Show solution
  1. The degree is the greatest exponent in the polynomial.
  2. The highest power is x^5.
  3. So the degree is 5.

Answer: 5

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