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:
- Identify the roots and critical points from the description.
- Use the critical points to help determine the coefficients.
Step-by-Step Solution
- Identify the roots and critical points:
- The curve passes through .
- The curve has a local maximum at .
- The curve has a local minimum at .
- The curve crosses the x-axis at .
The roots of the polynomial are and . Since it's a cubic polynomial, we can assume a double root at to account for the direction change.
So, the polynomial can be expressed as:
- **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
-
Suitable Grade Level
Grades 11-12
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