Math Problem Statement

Sketch the graph f(x) = 6x(x-1)^2(x-2)^3

Solution

To sketch the graph of the function f(x)=6x(x1)2(x2)3f(x) = 6x(x-1)^2(x-2)^3, let's first analyze its key features.

Step-by-Step Analysis:

  1. Zeros of the function: The function is factored, so we can determine the zeros by setting each factor equal to zero:

    • 6x=06x = 0x=0x = 0
    • (x1)2=0(x - 1)^2 = 0x=1x = 1
    • (x2)3=0(x - 2)^3 = 0x=2x = 2

    Therefore, the function has zeros at x=0x = 0, x=1x = 1, and x=2x = 2.

  2. Multiplicity of the roots:

    • At x=0x = 0, the factor xx has a multiplicity of 1 (linear).

    • At x=1x = 1, the factor (x1)2(x - 1)^2 has a multiplicity of 2 (even).

    • At x=2x = 2, the factor (x2)3(x - 2)^3 has a multiplicity of 3 (odd).

    • Multiplicity implications:

      • A root with odd multiplicity crosses the x-axis.
      • A root with even multiplicity touches the x-axis but doesn't cross.
  3. End behavior:

    • As x+x \to +\infty, the leading term is x6x^6, so f(x)+f(x) \to +\infty.
    • As xx \to -\infty, the leading term is still dominated by x6x^6, so f(x)+f(x) \to +\infty.
  4. Behavior at roots:

    • At x=0x = 0, the graph crosses the x-axis (since the multiplicity is odd).
    • At x=1x = 1, the graph touches the x-axis and turns back up (since the multiplicity is even).
    • At x=2x = 2, the graph crosses the x-axis with an "S" shape (since the multiplicity is odd).
  5. Shape of the graph:

    • The graph starts from positive infinity for large negative xx, comes down to zero at x=0x = 0, crosses the axis, touches the x-axis at x=1x = 1, and then crosses again at x=2x = 2 with a steep "S" shape.

Would you like me to plot the actual graph, or would you prefer a detailed numerical analysis for specific values of xx? Let me know if you have any questions!

Here are 5 related questions to expand your understanding:

  1. What is the significance of root multiplicity in determining how the graph behaves at a given point?
  2. How do you determine the end behavior of a polynomial function?
  3. How does changing the leading coefficient (e.g., the 66 in this case) affect the graph?
  4. How would the graph change if (x2)3(x-2)^3 had an even exponent instead of odd?
  5. How can you use the first and second derivatives to analyze the concavity of this function?

Tip: Always pay attention to the multiplicity of roots; even multiplicity means the graph touches but doesn't cross, while odd multiplicity means it crosses the x-axis.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Roots and Multiplicity
End Behavior

Formulas

f(x) = 6x(x-1)^2(x-2)^3
Multiplicity rules: even multiplicity - touch, odd multiplicity - cross

Theorems

Multiplicity Theorem
End Behavior of Polynomial Functions

Suitable Grade Level

Grades 10-12