Sketching Polynomial Graphs
A free Algebra II lesson from the “Polynomial Graphs” unit, with a worked example and practice problems including step-by-step solutions.
A polynomial sketch uses end behavior, x-intercepts, y-intercept, and multiplicity. The goal is not a perfect drawing; it is a graph that matches the important features.
What you'll learn
- Combine zeros and end behavior
- Estimate graph shape from factored form
- Use intercepts to sketch
Worked example
Problem. Sketch features for f(x) = (x - 2)(x + 1)^2.
- Zeros are 2 and -1.
- x - 2 has odd multiplicity, so the graph crosses at 2.
- x + 1 is squared, so the graph touches at -1.
Answer: crosses at 2, touches at -1, right end rises
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. For f(x) = (x - 3)(x + 2), the graph crosses at...
Choices: 3 and -2 · -3 and 2 · 3 only · -2 only
Show solution
- Both factors have odd multiplicity.
Answer: 3 and -2
2. For f(x) = (x - 4)^2, the graph...
Choices: Touches at 4 · Crosses at 4 · Touches at -4 · Has no zero
Show solution
- Even multiplicity touches.
Answer: Touches at 4
3. For f(x) = x(x - 5)^2, the graph touches at...
Choices: 5 · 0 · -5 · Both 0 and 5
Show solution
- The squared factor gives touching at 5.
Answer: 5
4. For f(x) = (x - 2)(x + 3), what is the y-intercept?
Show solution
- Evaluate f(0): (-2)(3) = -6.
Answer: -6
5. For f(x) = (x - 2)(x + 1), what is the y-intercept?
Show solution
- Set x = 0 for the y-intercept.
- f(0) = (0 - 2)(0 + 1).
- The y-intercept is -2.
Answer: -2
Practice this interactively with instant feedback and an AI tutor.
Practice Sketching Polynomial Graphs Take the free placement check