Math Problem Statement

The x y-coordinate plane is given. The curve enters the window in the fourth quadrant, goes up and right, passes through the point (3, −3), changes direction at the point (4, 0), goes down and right, changes direction at the approximate point (5.3, −1.2), goes up and right, crosses the x-axis at x = 6, and exits the window in the first quadrant. Find the polynomial function of degree 3 whose graph is shown in the figure.

Solution

To find the polynomial function of degree 3 given the described graph, we will follow these steps:

  1. Identify the roots and critical points from the description.
  2. Use the critical points to help determine the coefficients.

Step-by-Step Solution

  1. Identify the roots and critical points:
    • The curve passes through (3,3)(3, -3).
    • The curve has a local maximum at (4,0)(4, 0).
    • The curve has a local minimum at (5.3,1.2)(5.3, -1.2).
    • The curve crosses the x-axis at x=6x = 6.

The roots of the polynomial are x=3x = 3 and x=6x = 6. Since it's a cubic polynomial, we can assume a double root at x=6x = 6 to account for the direction change.

So, the polynomial can be expressed as: f(x)=a(x3)(x6)2f(x) = a(x - 3)(x - 6)^2

  1. **Determine the leading coefficient

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Roots and Critical Points
Cubic Equations

Formulas

Cubic polynomial formula: f(x) = a(x - r1)(x - r2)^2

Theorems

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Suitable Grade Level

Grades 11-12