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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

Why it matters: Sketches with the right zeros, end behavior, and crossing pattern let engineers and analysts read a polynomial's behavior without plotting every point. The leading term, the zeros, and the y-intercept are usually enough.

Worked example

Problem. Sketch features for f(x) = (x - 2)(x + 1)^2.

  1. Zeros are 2 and -1.
  2. x - 2 has odd multiplicity, so the graph crosses at 2.
  3. x + 1 is squared, so the graph touches at -1.

Answer: crosses at 2, touches at -1, right end rises

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
  1. Warm-up: First identify exactly what the question is asking: For f(x) = (x - 3)(x + 2), the graph crosses at...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Both factors have odd multiplicity.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. Warm-up: First identify exactly what the question is asking: For f(x) = (x - 4)^2, the graph...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Even multiplicity touches.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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
  1. Core Practice: First identify exactly what the question is asking: For f(x) = x(x - 5)^2, the graph touches at...
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. The squared factor gives touching at 5.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5

4. For f(x) = (x - 2)(x + 3), what is the y-intercept?

Show solution
  1. Challenge: First identify exactly what the question is asking: For f(x) = (x - 2)(x + 3), what is the y-intercept?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Evaluate f(0): (-2)(3) = -6.
  4. Check the result by substituting or estimating: the response should match -6 and make sense in the original problem.

Answer: -6

5. For f(x) = (x - 2)(x + 1), what is the y-intercept?

Show solution
  1. Review: First identify exactly what the question is asking: For f(x) = (x - 2)(x + 1), what is the y-intercept?
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Set x = 0 for the y-intercept.
  4. f(0) = (0 - 2)(0 + 1).
  5. The y-intercept is -2.
  6. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

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