Analyzing And Correcting Nick's Vertex Form Conversion Of P(x) = 17 + 42x - 7x²

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In this detailed analysis, we will dissect Nick's attempt to convert the quadratic function p(x) = 17 + 42x - 7x² into vertex form. Vertex form, a pivotal concept in quadratic functions, provides valuable insights into the parabola's key features, including its vertex, axis of symmetry, and maximum or minimum value. Nick's methodical approach, though containing a subtle error, offers an excellent opportunity to reinforce our understanding of completing the square and vertex form conversion. We will meticulously examine each step of Nick's work, pinpoint the discrepancy, and navigate the correct path to express the function in its accurate vertex form. By the end of this exploration, we will not only rectify Nick's work but also fortify our comprehension of the underlying principles governing quadratic transformations. This comprehensive analysis will serve as a powerful tool for students and educators alike, illuminating the nuances of quadratic functions and their graphical representations. Understanding vertex form unlocks a deeper understanding of quadratic behavior and its applications in various fields, making this exercise a cornerstone in mathematical proficiency. We will delve into the logic behind each manipulation, ensuring a clear grasp of the process rather than mere rote memorization. This journey into Nick's work is a journey into the heart of quadratic functions, promising a richer and more nuanced understanding of this fundamental mathematical concept.

Nick's Work Unveiled

Nick's methodical approach to transforming the quadratic function into vertex form is a commendable starting point. Let's meticulously examine each step, shedding light on the process and identifying any potential deviations from the correct path. Nick's initial steps are crucial in setting the stage for completing the square, a technique that forms the bedrock of vertex form conversion. A keen eye for detail is paramount as we navigate through the algebraic manipulations, ensuring the preservation of equality and the accurate application of mathematical principles. This step-by-step analysis will not only reveal the location of any error but also deepen our appreciation for the elegance and precision inherent in mathematical transformations. By closely scrutinizing each step, we gain not only a solution but also a more profound understanding of the underlying methodology. This approach fosters critical thinking and problem-solving skills, essential components of mathematical fluency. Understanding the rationale behind each step empowers us to tackle similar problems with confidence and clarity. The following breakdown presents Nick's work, ready for our critical evaluation.

  1. p(x) = -7x² + 42x + 17
  2. p(x) = -7(x² - 6x) + 17
  3. (-x/2)² = 9 ; p(x) = -7(x² - 6x + 9) + 17
  4. p(x) = -7(x - 3)² + 17

Step-by-Step Analysis of Nick's Solution

Step 1 A Rearrangement of Terms

In Nick's first step, p(x) = -7x² + 42x + 17, we observe a simple yet crucial rearrangement of terms. The purpose of this maneuver is to organize the quadratic expression in descending order of powers of x, a standard practice that facilitates subsequent steps in the vertex form conversion. This rearrangement, while seemingly trivial, brings the quadratic term (-7x²) and the linear term (42x) into proximity, setting the stage for factoring out the leading coefficient. This meticulous attention to order reflects a sound understanding of algebraic structure and its impact on mathematical operations. The rearrangement lays the foundation for the next critical step: factoring out the coefficient of the quadratic term. By arranging the terms in this manner, Nick demonstrates an awareness of the importance of structure in mathematical expressions. This seemingly simple step is a testament to the power of organization in simplifying complex problems. It underscores the principle that a well-structured problem is often half-solved. By starting with this rearrangement, Nick signals a clear and logical approach to the task at hand.

Step 2 Factoring the Leading Coefficient

Nick's second step, p(x) = -7(x² - 6x) + 17, demonstrates a key technique in preparing for completing the square: factoring out the leading coefficient (-7) from the quadratic and linear terms. This action isolates the x² and x terms within the parentheses, allowing us to focus on completing the square for the expression inside. The constant term (+17) remains outside the parentheses, as it will be adjusted later to maintain the equation's balance. Factoring out the leading coefficient is a strategic move, transforming the quadratic expression into a form more amenable to completing the square. This step highlights the importance of recognizing patterns and applying algebraic manipulations to simplify expressions. By factoring out -7, Nick has effectively created a quadratic expression within the parentheses that is easier to manipulate. This step showcases an understanding of the distributive property in reverse, a crucial skill in algebraic simplification. The process of factoring out the leading coefficient is a pivotal step in the transformation process.

Step 3 Completing the Square The Critical Point

Step 3, (-x/2)² = 9 ; p(x) = -7(x² - 6x + 9) + 17, is where we encounter a potential issue in Nick's work. While the calculation of the term needed to complete the square, (-6/2)² = 9, is correct, the subsequent addition of 9 inside the parentheses necessitates a corresponding adjustment outside the parentheses to maintain the equation's balance. Because the expression (x² - 6x + 9) is being multiplied by -7, adding 9 inside the parentheses is effectively adding -7 * 9 = -63 to the right side of the equation. Therefore, to compensate, we must add 63 outside the parentheses, not simply leave the constant term as +17. This is the crucial step where Nick's solution deviates from the correct path. Understanding the need to compensate for the addition within the parentheses is paramount in completing the square accurately. This step highlights the delicate balance inherent in algebraic manipulations and the importance of maintaining equality. The error in this step underscores the need for careful attention to detail when working with quadratic expressions. The ramifications of this error will propagate through the remaining steps, leading to an incorrect vertex form. Therefore, identifying and rectifying this error is essential for achieving the correct solution. The principle of maintaining balance in equations is a fundamental concept in algebra.

Step 4 Incorrectly Expressing in Vertex Form

Nick's final step, p(x) = -7(x - 3)² + 17, presents the function in what appears to be vertex form. However, due to the error in Step 3, this is not the correct vertex form representation of the original quadratic function. The expression (x² - 6x + 9) correctly factors into (x - 3)², but the constant term is incorrect because the compensation for completing the square was not properly accounted for. The vertex form of a quadratic function is p(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. In this case, the h value is correct (3), but the k value is incorrect due to the previous error. This step underscores the importance of verifying each step in the solution process, as an error early on can lead to a cascade of incorrect results. While the form of the expression resembles vertex form, the incorrect constant term renders it inaccurate. The vertex form provides valuable information about the parabola's position and orientation, making it essential to arrive at the correct representation. This step highlights the interconnectedness of mathematical operations and the need for precision in each step. The final expression is a consequence of the preceding steps, and an error in one step inevitably affects the outcome.

Correcting Nick's Work

To rectify Nick's work and arrive at the accurate vertex form, we must revisit Step 3 and make the necessary correction. As identified earlier, the addition of 9 inside the parentheses (multiplied by -7) necessitates adding 63 outside the parentheses to maintain the equation's balance. Let's walk through the corrected steps:

  1. p(x) = -7x² + 42x + 17 (No change)
  2. p(x) = -7(x² - 6x) + 17 (No change)
  3. (-6/2)² = 9 ; p(x) = -7(x² - 6x + 9) + 17 + 63 (Corrected Step)
  4. p(x) = -7(x - 3)² + 80 (Corrected Vertex Form)

By adding 63 to the constant term, we arrive at the correct vertex form: p(x) = -7(x - 3)² + 80. This form reveals that the vertex of the parabola is at the point (3, 80), and the parabola opens downwards due to the negative leading coefficient (-7). The corrected solution underscores the importance of meticulous attention to detail and the principle of maintaining balance in algebraic manipulations. This step-by-step correction reinforces the understanding of completing the square and its application in vertex form conversion. The corrected vertex form provides accurate information about the parabola's characteristics, enabling us to analyze its behavior and graph it correctly. This correction highlights the power of identifying and rectifying errors in mathematical problem-solving. The process of correcting Nick's work is a valuable learning experience, reinforcing our understanding of quadratic functions.

The Significance of Vertex Form

The vertex form, p(x) = a(x - h)² + k, is a powerful representation of a quadratic function, offering immediate insights into the parabola's key characteristics. The vertex, (h, k), represents the point where the parabola changes direction, either reaching its maximum or minimum value. The sign of the leading coefficient, a, determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The axis of symmetry is a vertical line passing through the vertex, with the equation x = h. Understanding the vertex form empowers us to quickly sketch the graph of a quadratic function, identify its maximum or minimum value, and solve related optimization problems. The vertex form also facilitates the conversion between different forms of a quadratic function, such as standard form and factored form. This versatile representation is a cornerstone in the study of quadratic functions and their applications in various fields, including physics, engineering, and economics. Mastering vertex form unlocks a deeper understanding of quadratic behavior and its graphical representation. The ability to interpret and manipulate vertex form is a crucial skill in mathematical proficiency. Vertex form provides a concise and informative way to represent quadratic functions, making it an invaluable tool in mathematical analysis.

Conclusion

Through a careful examination of Nick's attempt to convert the quadratic function into vertex form, we've not only identified and corrected an error but also reinforced our understanding of the underlying mathematical principles. The importance of maintaining balance in algebraic manipulations, the significance of completing the square accurately, and the power of vertex form in revealing a parabola's characteristics have all been highlighted. This exercise serves as a valuable reminder of the need for meticulous attention to detail and a deep understanding of mathematical concepts. By dissecting Nick's work, we've gained a more nuanced appreciation for the elegance and precision inherent in quadratic functions. The journey from identifying an error to arriving at the correct solution is a testament to the power of critical thinking and problem-solving in mathematics. This exploration of Nick's work underscores the importance of practice and perseverance in mastering mathematical concepts. The corrected vertex form provides a clear and accurate representation of the quadratic function, enabling us to analyze its behavior and applications. This comprehensive analysis has not only rectified an error but also deepened our understanding of quadratic functions and their significance.