Finding the Equation of a Degree 6 Polynomial Function

TLDR; Finding the equation of a degree 6 polynomial function from the graph and determining the value of 'A' to represent the graph.

💡 Finding the Equation

The goal is to find an equation for the graph of a degree 6 polynomial function and leave it in factored form, considering at most 6 real rational zeros.

By analyzing the x-intercepts of the graph, it's possible to identify the number of real rational zeros or roots and their multiplicities based on how the function behaves when crossing or touching the x-axis.

The zeros at (-2, +5) touch but don't cross the x-axis, indicating an even multiplicity, leading to the assumption that both have a multiplicity of 2, considering the four x-intercepts.

🔍 Using the Factored Form

To find the equation of the function, the factored form of a polynomial is used, with 'A' as a constant and the roots or zeros denoted as R sub 1, R sub 2, R sub 3, and so on.

The value of 'A' is determined using the factored form of the polynomial and one more point on the function, such as the y-intercept at (0, -30).

📈 Finding the Factors

Based on the identified zeros or roots, the factors of the polynomial function are found, taking into account their multiplicities.

This involves creating factors for each zero, considering their multiplicity, and visualizing how the graph behaves.

⚖️ Solving for 'A'

Substituting the value of 'A' is determined by using the y-intercept point (0, -30) to solve for 'A' in the function equation, resulting in the value of 'A' being equal to +1/10.

🧮 Using All Information

With the value of 'A' known, all the necessary information is available to find a function to represent the graph, using the determined value of 'A' and rearranging the factors in the equation accordingly.

📚 Conclusion

The process of determining the equation of a degree 6 polynomial function from the graph and finding the value of 'A' provides a helpful method for representing complex functions and understanding their behavior.