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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

Why it matters: Polynomial models describe smooth turning behavior in design, approximation, data fitting, and STEM modeling.

Worked example

Problem. For p(x) = (x - 3)(x + 2), which x-values should be marked as zeros?

  1. Worked Example: First identify exactly what the question is asking: For p(x) = (x - 3)(x + 2), which x-values should be marked as zeros?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set each factor equal to 0.
  4. x - 3 = 0 gives x = 3.
  5. x + 2 = 0 gives x = -2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 3 and x = -2

Practice problems

1. For p(x) = (x - 3)(x + 2), which x-values should be marked as zeros?

Choices: x = 3 and x = -2 · x = -3 and x = 2 · x = 3 and x = 2 · x = -3 and x = -2

Show solution
  1. Warm-up: First identify exactly what the question is asking: For p(x) = (x - 3)(x + 2), which x-values should be marked as zeros?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Set each factor equal to 0.
  4. x - 3 = 0 gives x = 3.
  5. x + 2 = 0 gives x = -2.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x = 3 and x = -2

2. How many distinct x-intercepts does p(x) = (x - 4)(x + 3)(x - 6) have?

Show solution
  1. Each factor gives one zero: x = 4, x = -3, x = 6.
  2. List the distinct values: 4, -3, 6 are all different.
  3. So there are 3 distinct x-intercepts.

Answer: 3

3. p(x) = (x - 2)(x + 4)^2 has a repeated factor. How many distinct x-intercepts does its graph have?

Show solution
  1. The factors are x - 2 and x + 4 (squared).
  2. Distinct zeros are x = 2 and x = -4 — the repeat does not add a new location.
  3. So there are 2 distinct x-intercepts.

Answer: 2

4. A polynomial has degree 7. What is the maximum number of turning points its graph can have?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A polynomial has degree 7. What is the maximum number of turning points its graph can have?
  2. For polynomial work, use degree, leading terms, and like terms to keep the expression organized.
  3. A degree-n graph turns at most n - 1 times.
  4. Here n = 7, so the maximum is 7 - 1.
  5. That is 6 turning points.
  6. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. Find the y-intercept of p(x) = (x - 4)(x + 2). Enter p(0).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the y-intercept of p(x) = (x - 4)(x + 2). Enter p(0).
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. The y-intercept is p(0).
  4. p(0) = (0 - 4)(0 + 2) = (-4)(2).
  5. So p(0) = -8.
  6. Check the result by substituting or estimating: the response should match -8 and make sense in the original problem.

Answer: -8

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