Finding The Quadratic Function Equation With Vertex (2,-25) And X-intercept (7,0)

In the realm of mathematics, quadratic functions hold a prominent position, renowned for their elegant parabolic curves and wide-ranging applications. These functions, often expressed in the form f(x) = ax² + bx + c, possess a unique vertex, a point where the parabola attains its minimum or maximum value, and may also intersect the x-axis at one or two points, known as x-intercepts. In this article, we embark on a journey to unravel the equation of a specific quadratic function, one that boasts a vertex at (2,-25) and an x-intercept at (7,0). Our quest will involve a blend of algebraic manipulation and insightful utilization of the vertex form of a quadratic equation.

Leveraging the Vertex Form: A Gateway to the Equation

The vertex form of a quadratic equation serves as our cornerstone in this endeavor. This form, expressed as f(x) = a(x - h)² + k, elegantly encapsulates the vertex coordinates (h, k), providing a direct link between the equation and the parabola's pivotal point. In our case, the vertex is given as (2, -25), allowing us to readily substitute these values into the vertex form, yielding f(x) = a(x - 2)² - 25. The coefficient a remains an enigma, a scaling factor that dictates the parabola's concavity and width. To unveil the value of a, we turn to the provided x-intercept, (7, 0), a point where the function's output gracefully intersects the x-axis.

Decoding the X-intercept: A Key to Unveiling 'a'

The x-intercept, (7, 0), signifies that when x = 7, the function's value, f(x), gracefully diminishes to zero. This harmonious relationship provides a crucial piece of the puzzle, allowing us to substitute these values into our partially unveiled equation, f(x) = a(x - 2)² - 25. By replacing f(x) with 0 and x with 7, we arrive at the equation 0 = a(7 - 2)² - 25. This equation, a beacon of algebraic simplicity, holds the key to unraveling the value of a. Through judicious manipulation, we add 25 to both sides, resulting in 25 = a(5)², which gracefully simplifies to 25 = 25a. Dividing both sides by 25, we unveil the value of a, a satisfying 1. With a now illuminated, our equation takes on a more complete form: f(x) = 1(x - 2)² - 25.

Expanding and Simplifying: Revealing the Standard Form

Our quest for the equation of the quadratic function now leads us to expand and simplify the expression f(x) = 1(x - 2)² - 25. Expanding the squared term, (x - 2)², we obtain x² - 4x + 4. Substituting this expansion back into our equation, we arrive at f(x) = 1(x² - 4x + 4) - 25. Distributing the 1, which gracefully leaves the expression unchanged, we have f(x) = x² - 4x + 4 - 25. Finally, combining the constant terms, 4 and -25, we unveil the standard form of our quadratic equation: f(x) = x² - 4x - 21. This equation, a testament to our algebraic journey, elegantly captures the essence of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0).

Factoring to Uncover the Roots: An Alternative Perspective

The standard form of our quadratic equation, f(x) = x² - 4x - 21, provides an alternative pathway to understanding the function's behavior. By factoring this equation, we can unearth the roots, the x-values where the function gracefully intersects the x-axis. Factoring x² - 4x - 21, we seek two numbers that multiply to -21 and add to -4. Through careful consideration, we identify -7 and 3 as the ideal candidates. Thus, we can rewrite the equation as f(x) = (x - 7)(x + 3). This factored form elegantly reveals the roots of the function: x = 7 and x = -3. The x-intercept (7, 0), which served as our initial clue, is now reaffirmed, while the newly discovered root, x = -3, unveils another point where the parabola gracefully crosses the x-axis.

Connecting the Dots: Roots, Vertex, and the Parabola's Embrace

The roots of the quadratic function, x = 7 and x = -3, provide valuable insights into the parabola's shape and position. The parabola gracefully intersects the x-axis at these two points, creating a visual representation of the function's behavior. The vertex, (2, -25), sits majestically at the parabola's turning point, marking either the minimum or maximum value of the function. The axis of symmetry, a vertical line that gracefully divides the parabola into two mirror images, passes through the vertex. In our case, the axis of symmetry is the line x = 2. The roots, the vertex, and the axis of symmetry work in harmonious concert, painting a complete picture of the quadratic function's parabolic dance.

Choosing the Correct Option: A Culmination of Our Journey

Having meticulously unraveled the equation of the quadratic function, we now stand poised to select the correct option from the given choices. Our derived equation, f(x) = (x - 7)(x + 3), aligns perfectly with option A: f(x) = (x + 3)(x - 7). The commutative property of multiplication assures us that the order of the factors does not alter the result. Therefore, option A emerges as the definitive answer, a testament to our algebraic prowess and insightful exploration.

Conclusion: A Triumph of Mathematical Exploration

Our journey to uncover the equation of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0) has been a resounding success. Through the strategic application of the vertex form, the insightful utilization of the x-intercept, and the elegant dance of algebraic manipulation, we have unveiled the equation f(x) = (x - 7)(x + 3). This equation, a culmination of our efforts, stands as a testament to the power of mathematical exploration and the beauty of quadratic functions.

In summary, quadratic functions are a crucial part of mathematics, and understanding how to derive their equations from given information is a valuable skill. We started by recognizing the vertex form of a quadratic equation, which allowed us to easily incorporate the given vertex coordinates. By using the x-intercept, we were able to solve for the unknown coefficient and ultimately express the function in both standard and factored forms. This comprehensive approach not only provided the answer but also deepened our understanding of the relationship between a quadratic function's key features and its equation. Remember, the vertex form, x-intercepts, and roots are all interconnected, offering different perspectives on the same underlying function. Mastery of these concepts will undoubtedly strengthen your mathematical toolkit and empower you to tackle a wide range of problems involving parabolas and quadratic equations.

Understanding the relationship between the vertex, x-intercepts, and the equation of a quadratic function is crucial for solving many mathematical problems. The ability to move between different forms of the equation (vertex form, standard form, factored form) allows for a flexible and comprehensive approach to problem-solving. This detailed exploration of the problem, from the initial vertex form to the final factored form, illustrates the power and beauty of mathematical reasoning. So, embrace the challenge, practice your skills, and continue to explore the fascinating world of mathematics! Understanding quadratic functions opens doors to more advanced mathematical concepts, making it a worthwhile endeavor for any student. Keep learning, keep practicing, and keep pushing the boundaries of your mathematical knowledge!

By mastering these concepts, you'll gain a strong foundation for tackling more complex mathematical problems and appreciating the elegance and power of quadratic functions. Remember to practice regularly and explore different types of problems to solidify your understanding. Mathematics is a journey of discovery, and each problem solved is a step forward on that journey. This problem, in particular, highlights the importance of connecting different pieces of information to arrive at a solution, a skill that is valuable not only in mathematics but in many areas of life. So, keep practicing, keep exploring, and keep the spirit of mathematical inquiry alive!