Finding Quadratic Equation From Vertex And X-intercept

At the heart of algebra lies the quadratic function, a powerful tool for modeling curves and parabolic shapes. Characterized by its general form, f(x) = ax² + bx + c, a quadratic function unveils its secrets through its key features: the vertex, the axis of symmetry, and the intercepts. Mastering the art of deciphering these features allows us to unlock the equation that governs the function's behavior.

Delving into the Vertex Form: The vertex form of a quadratic equation, f(x) = a(x - h)² + k, presents a unique perspective, directly highlighting the vertex of the parabola at the point (h, k). The vertex, the parabola's turning point, serves as a cornerstone for understanding the function's behavior. The coefficient a plays a crucial role, dictating the parabola's concavity – whether it opens upwards (a > 0) or downwards (a < 0) – and its vertical stretch or compression. This form provides a direct pathway to identifying the vertex and understanding the parabola's orientation.

Unraveling the X-Intercepts: The x-intercepts, the points where the parabola intersects the x-axis, hold vital clues about the function's roots or zeros. These points, where f(x) = 0, represent the solutions to the quadratic equation. Determining the x-intercepts involves setting the quadratic function equal to zero and solving for x. Techniques such as factoring, completing the square, or employing the quadratic formula come into play to unveil these critical points. Understanding the x-intercepts enriches our comprehension of the function's behavior and its relationship with the x-axis.

The Interplay of Vertex and X-Intercepts: The vertex and x-intercepts form a harmonious duo, providing a comprehensive view of the quadratic function's trajectory. The vertex anchors the parabola, while the x-intercepts mark its intersections with the x-axis. Armed with this information, we can sketch the parabola's graph, predict its behavior, and construct the equation that governs its path. The vertex form, coupled with the knowledge of x-intercepts, empowers us to decipher the secrets of quadratic functions and apply them to a myriad of real-world scenarios.

Determining the Quadratic Function's Equation

Let's embark on a journey to determine the equation of a quadratic function, armed with the knowledge of its vertex and an x-intercept. The vertex, a crucial landmark on the parabola, provides the coordinates (h, k), while the x-intercept, a point where the parabola intersects the x-axis, offers a solution to the quadratic equation. Combining these pieces of information, we can construct the equation that governs the function's behavior.

Harnessing the Vertex Form: The vertex form, f(x) = a(x - h)² + k, emerges as our primary tool. The vertex coordinates (h, k) directly plug into this form, narrowing down the possibilities. The remaining unknown, the coefficient a, holds the key to the parabola's concavity and vertical stretch. The x-intercept, a point on the parabola, steps in to unveil the value of a. By substituting the x-intercept's coordinates into the partially formed equation, we create an equation with a as the sole unknown. Solving for a completes the vertex form, providing a concise representation of the quadratic function.

Decoding the X-Intercept: The x-intercept, where f(x) = 0, acts as a strategic entry point. By substituting the x-intercept's coordinates into the vertex form, we transform the equation into a solvable puzzle. The y-coordinate, being zero at the x-intercept, simplifies the equation, leaving a as the sole unknown. Isolating a reveals its value, completing the vertex form and fully defining the quadratic function. This strategic maneuver showcases the x-intercept's power in deciphering the function's equation.

Constructing the Equation: With the vertex and an x-intercept in hand, we embark on a step-by-step construction of the quadratic function's equation. The vertex form, f(x) = a(x - h)² + k, serves as our blueprint. The vertex coordinates (h, k) seamlessly integrate into this form, while the x-intercept's coordinates act as a decoder for the coefficient a. By substituting the x-intercept's values and solving for a, we unveil the final piece of the puzzle. The complete vertex form, now adorned with the value of a, stands as the unique equation that governs the quadratic function's behavior, paving the way for further analysis and applications.

Step-by-Step Solution

Let's apply our knowledge to solve the problem at hand: determining the equation of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0). We'll follow a step-by-step approach, harnessing the power of the vertex form and the strategic use of the x-intercept.

1. Embracing the Vertex Form: We begin by embracing the vertex form, f(x) = a(x - h)² + k, the cornerstone of our solution. The vertex coordinates, (2, -25), seamlessly integrate into this form, replacing h with 2 and k with -25. This substitution yields f(x) = a(x - 2)² - 25, a partially formed equation that awaits the unveiling of the coefficient a.

2. Decoding the X-Intercept: The x-intercept, (7, 0), emerges as our strategic decoder. Substituting these coordinates into the partially formed equation, we replace x with 7 and f(x) with 0. This substitution transforms the equation into 0 = a(7 - 2)² - 25, a solvable puzzle with a as the sole unknown. The x-intercept's power lies in its ability to convert the function's equation into a solvable expression, paving the way for us to unveil the value of a.

3. Unveiling the Coefficient a: With the equation 0 = a(7 - 2)² - 25 in our grasp, we embark on the quest to isolate a. Simplifying the equation, we have 0 = 25a - 25. Adding 25 to both sides, we get 25 = 25a. Dividing both sides by 25, we reveal the value of a as 1. The coefficient a, now unveiled, completes the vertex form and brings us closer to the final equation.

4. Constructing the Equation: With the value of a determined, we complete the vertex form. Substituting a = 1 into f(x) = a(x - 2)² - 25, we arrive at f(x) = (x - 2)² - 25. This equation, in vertex form, governs the quadratic function's behavior. To align with the answer choices, we expand the equation: f(x) = (x² - 4x + 4) - 25, which simplifies to f(x) = x² - 4x - 21. This equation, now in standard form, reveals the function's coefficients and provides a different perspective on its behavior.

5. Factoring for Clarity: To further refine our understanding and match the answer choices, we factor the equation f(x) = x² - 4x - 21. Factoring the quadratic expression, we obtain f(x) = (x - 7)(x + 3). This factored form elegantly showcases the function's x-intercepts, the roots of the equation, and provides a clear representation of the parabola's behavior. The solution, f(x) = (x - 7)(x + 3), aligns perfectly with the structure of the answer choices, solidifying our understanding of the quadratic function and its equation.

Analyzing the Options

Now, let's scrutinize the given options, comparing them with our derived equation, f(x) = (x - 7)(x + 3), to pinpoint the correct answer.

Option A: f(x) = (x - 2)(x - 7): This option immediately raises a red flag. While it correctly identifies the x-intercept at (7, 0), it fails to incorporate the vertex at (2, -25). The factor (x - 2) might seem related to the vertex's x-coordinate, but it doesn't account for the parabola's vertical shift or the y-coordinate of the vertex. This option, therefore, falls short of accurately representing the quadratic function.

Option B: f(x) = (x + 2)(x + 7): This option veers further away from the correct equation. It not only fails to account for the vertex but also incorrectly represents the x-intercepts. The factors (x + 2) and (x + 7) would imply x-intercepts at (-2, 0) and (-7, 0), respectively, contradicting the given x-intercept at (7, 0). This option demonstrates a misunderstanding of both the vertex and the x-intercepts.

Option C: f(x) = (x - 3)(x + 7): This option presents a mix of correct and incorrect elements. It accurately captures the x-intercept at (7, 0) through the factor (x - 7). However, the factor (x + 3) suggests an x-intercept at (-3, 0), which doesn't align with the given information. Furthermore, this option doesn't explicitly incorporate the vertex coordinates, making it an incomplete representation of the quadratic function.

Option D: f(x) = (x + 3)(x - 7): This option emerges as the victor, perfectly aligning with our derived equation, f(x) = (x - 7)(x + 3). The factors (x + 3) and (x - 7) accurately represent the x-intercepts at (-3, 0) and (7, 0), respectively. While this form doesn't explicitly showcase the vertex, it implicitly incorporates its information through the factored form. Expanding this option would lead back to the standard form, f(x) = x² - 4x - 21, which we derived earlier. This option stands as the correct answer, capturing the essence of the quadratic function and its key features.

Conclusion

The journey to decipher the equation of a quadratic function, armed with the vertex and an x-intercept, highlights the power of strategic problem-solving. The vertex form, f(x) = a(x - h)² + k, serves as our compass, guiding us towards the solution. The x-intercept, acting as a decoder, unveils the value of the coefficient a. By carefully constructing the equation and analyzing the options, we arrive at the correct answer, f(x) = (x + 3)(x - 7). This exploration reinforces our understanding of quadratic functions and their equations, paving the way for further mathematical adventures.