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
- Use degree and leading coefficient
- Predict end behavior
- Connect algebra to graph shape
Worked example
Problem. Describe the end behavior of y = -2x^4 + 3x - 1.
- The degree is even.
- The leading coefficient is negative.
- Even degree with negative leading coefficient means both ends go down.
Answer: both ends down
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
- Warm-up: First identify exactly what the question is asking: The graph of y = x^4 has ends that...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Even degree and positive leading coefficient.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: The graph of y = -x^3 has ends that...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Odd degree and negative leading coefficient.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Core Practice: First identify exactly what the question is asking: For y = 5x^6 - x + 1, the end behavior is...
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Even degree with positive leading coefficient.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Both rise
4. For y = -3x^5 + 2x^2, the right end...
Choices: Falls · Rises · Stays flat · Crosses twice
Show solution
- Challenge: First identify exactly what the question is asking: For y = -3x^5 + 2x^2, the right end...
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Odd degree with negative leading coefficient falls right.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Falls
5. What is the degree of 4x^5 - 2x^2 + 1?
Show solution
- Review: First identify exactly what the question is asking: What is the degree of 4x^5 - 2x^2 + 1?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- The degree is the greatest exponent in the polynomial.
- The highest power is x^5.
- So the degree is 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
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