Graphing Polynomial Functions
A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.
Polynomial graphing is a feature checklist: end behavior, zeros, multiplicities, and a few plotted values. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Sketch polynomial graphs from degree, leading term, zeros, and multiplicity
- Use graphing polynomial functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. For p(x) = (x + 2)(x - 1)^2, which zeros should be marked?
- Worked Example: First identify exactly what the question is asking: For p(x) = (x + 2)(x - 1)^2, which zeros should be marked?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Set each factor equal to 0.
- x + 2 gives x = -2.
- x - 1 gives x = 1.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x = -2 and x = 1
Practice problems
1. For p(x) = (x + 2)(x - 1)^2, which zeros should be marked?
Choices: x = -2 and x = 1 · x = 2 and x = -1 · x = -2 only · x = 1 only
Show solution
- Warm-up: First identify exactly what the question is asking: For p(x) = (x + 2)(x - 1)^2, which zeros should be marked?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Set each factor equal to 0.
- x + 2 gives x = -2.
- x - 1 gives x = 1.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x = -2 and x = 1
2. For p(x) = (x + 2)(x - 1)^2, which zeros should be marked? (variation 2)
Choices: x = -2 and x = 1 · x = 2 and x = -1 · x = -2 only · x = 1 only
Show solution
- Warm-up: First identify exactly what the question is asking: For p(x) = (x + 2)(x - 1)^2, which zeros should be marked?
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Set each factor equal to 0.
- x + 2 gives x = -2.
- x - 1 gives x = 1.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x = -2 and x = 1
3. A polynomial sketch should usually start with:
Choices: end behavior and zeros · only the y-axis label · a logarithm table · a denominator restriction
Show solution
- End behavior gives the broad direction.
- Zeros place x-axis interactions.
- Then multiplicity and points refine the sketch.
Answer: end behavior and zeros
4. A polynomial sketch should usually start with: (variation 2)
Choices: end behavior and zeros · only the y-axis label · a logarithm table · a denominator restriction
Show solution
- End behavior gives the broad direction.
- Zeros place x-axis interactions.
- Then multiplicity and points refine the sketch.
Answer: end behavior and zeros
5. If a cubic has positive leading coefficient and zeros -1, 2, and 4, the right end goes:
Choices: up · down · left · nowhere
Show solution
- A cubic is odd degree.
- Positive leading coefficient means right end rises.
- The zeros locate crossings but do not change that right-end direction.
Answer: up
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