Math Problem Statement

Here is the graph of a polynomial function with degree 4. Select all of the statements that are true about the function.

Solution

Let's analyze the graph and the corresponding statements about the degree 4 polynomial function.

  1. Statement A: The leading coefficient is positive.

    • The end behavior of the graph shows that as xx \to -\infty, yy \to \infty, and as xx \to \infty, yy \to \infty. This indicates that the leading coefficient is positive for a degree 4 polynomial.
    • This statement is true.
  2. Statement B: The constant term is negative.

    • The constant term is the yy-intercept of the polynomial, which occurs when x=0x = 0. From the graph, the yy-intercept is positive (above the x-axis).
    • This statement is false.
  3. Statement C: It has 2 relative maximums.

    • Looking at the graph, there are two points where the function has relative maximums: one near x=6x = -6 and another around x=3x = 3.
    • This statement is true.
  4. Statement D: It has 4 linear factors.

    • A degree 4 polynomial typically has four factors if all the roots are real. This graph crosses the x-axis four times, suggesting four real roots.
    • This statement is true.
  5. Statement E: One of the factors is (x1)(x - 1).

    • Looking at the x-intercepts, the function crosses the x-axis at around x=1x = 1, so (x1)(x - 1) could be one of the factors.
    • This statement is true.
  6. Statement F: One of the zeros is x=2x = 2.

    • From the graph, it does not seem that the function crosses the x-axis at x=2x = 2; it crosses at x=1x = 1 and around x=4x = 4.
    • This statement is false.
  7. Statement G: There is a relative minimum between x=1x = 1 and x=3x = 3.

    • Indeed, there is a relative minimum around x=2x = 2.
    • This statement is true.

Correct Statements:

  • A, C, D, E, G

Would you like any further clarifications or details on these? Here are 5 related questions for deeper understanding:

  1. What does the behavior of a degree 4 polynomial tell us about its graph?
  2. How can we determine the number of real roots from a polynomial graph?
  3. What is the significance of the leading coefficient in determining end behavior?
  4. How do relative maximum and minimum points help us analyze a graph?
  5. Why does the constant term correspond to the yy-intercept?

Tip: Always check the end behavior of a polynomial by looking at the degree and sign of the leading coefficient—it gives important clues about the function's general shape.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Degree of Polynomials
End Behavior
Relative Maximum and Minimum
Roots of Polynomial Equations

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12