Math Problem Statement

Identify the equation of the polynomial function based on the graph.

Solution

The graph provided shows a polynomial function, and the problem asks to identify its equation from the given options. Based on the graph:

  1. Roots of the Polynomial:
    The graph crosses or touches the x-axis at three points:

    • x=1x = 1
    • x=4x = 4
    • x=6x = 6
  2. Behavior at Each Root:

    • At x=1x = 1, the graph touches the x-axis and turns around, indicating a double root (multiplicity 2).
    • At x=4x = 4, the graph crosses the x-axis with a linear (straight-through) behavior, indicating a single root (multiplicity 1).
    • At x=6x = 6, the graph touches the x-axis and turns around, indicating another double root (multiplicity 2).
  3. Equation of the Polynomial:
    Based on the roots and their multiplicities, the equation can be written as:
    y=(x1)2(x4)1(x6)2y = (x - 1)^2 (x - 4)^1 (x - 6)^2

  4. Answer:
    The correct equation from the options is:
    y=(x1)2(x4)1(x6)2y = (x - 1)^2 (x - 4)^1 (x - 6)^2

Would you like further details about the reasoning? Here are some related questions to deepen your understanding:


5 Related Questions:

  1. How can the multiplicity of a root be determined by looking at a graph?
  2. Why do double roots cause the graph to touch but not cross the x-axis?
  3. How can we write the equation of a polynomial if the graph shows a triple root at a certain point?
  4. What is the general form of a polynomial function with nn distinct roots and their multiplicities?
  5. How does the leading coefficient affect the end behavior of a polynomial graph?

Tip:

When analyzing polynomial graphs, always check the behavior at the x-intercepts to determine the root multiplicities.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Roots and Multiplicity
Graphing Polynomials

Formulas

y = (x - r1)^m1 (x - r2)^m2 ... (x - rn)^mn, where r is the root and m is the multiplicity

Theorems

Root-Multiplicity Behavior Theorem

Suitable Grade Level

Grades 10-12