Determining the Equation of a Quadratic Function with Given Roots and a Point

Introduction: In this article, we will explore the process of determining the equation of a quadratic function when given its roots and a point it passes through. We will use the method of factored form and the point to derive the equation in standard form. This method provides a step-by-step guide that is both clear and comprehensive, ensuring a deep understanding of the underlying principles.

Step 1: Forming the Factored Form of the Quadratic Function

Given the roots of a quadratic function, the most straightforward form to start with is the factored form. For the roots r1 -2 and r2 5, the factored form of the quadratic equation can be expressed as:

(f(x) a(x 2)(x - 5))

Step 2: Expanding the Factored Form

Next, we expand the factored form using the distributive property (FOIL - First, Outer, Inner, Last). This step provides us with the expanded form in terms of x with a coefficient a to be determined:

(f(x) a(x 2)(x - 5) a(x^2 - 3x - 10))

Step 3: Using the Given Point to Find the Value of a

Now, we use the given point (1, 6) to find the value of a. Substituting x 1 and f(1) 6 into the equation:

(6 a(1^2 - 3(1) - 10) a(1 - 3 - 10) a(-12))

(6 -12a)

(a -frac{1}{2})

Step 4: Writing the Final Equation in Standard Form

With the value of a determined, we can now write the quadratic function in standard form:

(f(x) -frac{1}{2}(x^2 - 3x - 10) -frac{1}{2}x^2 frac{3}{2}x 5)

Conclusion

In summary, the process of determining the equation of a quadratic function with given roots and a point involves forming the factored form, expanding it, substituting the given point to find the coefficient, and then writing the final equation in standard form. This method ensures a clear and logical sequence, which can be applied to any similar problem with roots and a given point.

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Resources

Additional resources, such as algebra textbooks and online tutorials, offer step-by-step guidance and practice problems to help deepen your understanding of algebraic concepts.

Important Notes

Throughout the process, it is crucial to pay attention to the sign changes and the importance of accurate substitution. Mistakes can often be traced to a simple arithmetic or sign error, so take your time and double-check your work.