Sandra's Vertex Form Conversion Error Analysis Of Quadratic Function Transformation
Sandra embarked on the task of converting the quadratic function p(x) = 30x + 5x^2 into vertex form. Vertex form, represented as p(x) = a(x - h)^2 + k, provides valuable insights into the parabola's vertex (h, k) and its overall shape. Sandra's step-by-step approach offers a glimpse into the process, but let's dissect her work to ensure accuracy and understanding.
Step 1: Rearranging the Terms
Sandra begins by rearranging the terms of the quadratic function, presenting it in the standard form of a quadratic equation. This rearrangement, changing p(x) = 30x + 5x^2 to p(x) = 5x^2 + 30x, is a crucial initial step. By placing the quadratic term first, followed by the linear term, she sets the stage for subsequent manipulations required to complete the square. This step highlights the importance of recognizing the structure of a quadratic expression and preparing it for further analysis. The standard form ax^2 + bx + c allows for easier identification of coefficients and facilitates the process of converting to vertex form. This seemingly simple rearrangement is a foundational element in understanding and manipulating quadratic functions effectively.
Step 2: Factoring out the Leading Coefficient
In the second step, Sandra factors out the leading coefficient, 5, from the terms involving x. This is a pivotal move in the process of completing the square. By extracting the 5, she isolates the quadratic and linear terms within the parentheses, making it easier to manipulate them into a perfect square trinomial. The resulting expression, p(x) = 5(x^2 + 6x), showcases the isolation of the key components needed for completing the square. This step underscores the significance of strategic factoring in algebraic manipulations. It demonstrates how isolating specific terms can simplify complex expressions and pave the way for further transformations. Factoring out the leading coefficient is a common technique used to rewrite quadratic expressions and ultimately solve quadratic equations or, as in this case, convert them into vertex form.
Step 3: Completing the Square An Erroneous Calculation
Sandra's third step introduces a potential misstep. She attempts to calculate the value needed to complete the square. The correct procedure involves taking half of the coefficient of the x term (which is 6), squaring it, and adding the result inside the parentheses. However, Sandra's calculation, (4/2)^2 = 9, seems to stem from an incorrect number. It should be (6/2)^2 rather than (4/2)^2. This error, while seemingly minor, has significant implications for the accuracy of the final vertex form. The process of completing the square relies on precisely determining the constant term that transforms the quadratic expression into a perfect square trinomial. An incorrect calculation at this stage will propagate through the remaining steps, leading to an inaccurate representation of the function in vertex form. Identifying and rectifying this error is crucial for achieving a correct conversion.
Step 4: Completing the Square The Correct Application
Building upon the (incorrect) result from Step 3, Sandra proceeds to add and subtract a value within the expression to complete the square. She adds 5(9) inside the parentheses and subtracts 5(9) outside to maintain the equation's balance. This step demonstrates the core principle of completing the square: adding and subtracting the same value ensures that the function's value remains unchanged while allowing for the creation of a perfect square trinomial. The expression p(x) = 5(x^2 + 6x + 9) - 5(9) showcases this manipulation. The key is to recognize that adding a constant inside the parentheses, which is being multiplied by the factored-out coefficient 5, necessitates subtracting an equivalent amount outside the parentheses to preserve the function's integrity. This step highlights the careful balancing act required when manipulating algebraic expressions to achieve a desired form.
Step 5: Expressing in Vertex Form
In the final step, Sandra expresses the quadratic function in vertex form. She correctly identifies the perfect square trinomial within the parentheses and factors it as (x + 3)^2. This is a crucial step in achieving the vertex form, which reveals the vertex of the parabola. The resulting expression, p(x) = 5(x + 3)^2 - 45, represents the function in vertex form, where the vertex is at (-3, -45). However, due to the error in Step 3, this vertex is not the true vertex of the original function. Despite the error, this step demonstrates the power of completing the square to transform a quadratic function into a form that readily reveals key features, such as the vertex. Vertex form provides a clear understanding of the parabola's position and orientation in the coordinate plane.
Sandra's function, as she derived it, is p(x) = 5(x + 3)^2 - 45. This represents a parabola that opens upwards (since the leading coefficient is positive) and has a vertex at (-3, -45). However, due to the error in calculating the value to complete the square, this is not the correct vertex form of the original function. The critical error lies in Step 3, where Sandra incorrectly calculated the value needed to complete the square. Instead of (6/2)^2 = 9, she used (4/2)^2 = 9, which is a coincidence, but the process that lead to that was not the right one. This error propagated through the remaining steps, leading to an incorrect vertex form.
Correcting Sandra's Work
To correct Sandra's work, we need to revisit Step 3 and perform the correct calculation for completing the square. The coefficient of the x term is 6. Half of 6 is 3, and squaring 3 gives us 9. Therefore, the correct value to add and subtract is 9. Here's the corrected process:
- p(x) = 5x^2 + 30x
- p(x) = 5(x^2 + 6x)
- (6/2)^2 = 9*
- p(x) = 5(x^2 + 6x + 9) - 5(9)
- p(x) = 5(x + 3)^2 - 45
Notice that even with the corrected calculation, the final vertex form remains the same. This highlights the importance of understanding the underlying concepts and carefully executing each step. While the numerical answer might sometimes appear correct due to coincidences, a flawed process can mask underlying errors.
The Correct Vertex Form and its Implications
The corrected vertex form, p(x) = 5(x + 3)^2 - 45, tells us that the parabola has a vertex at (-3, -45) and opens upwards. This information is crucial for understanding the function's behavior, finding its minimum value, and graphing it accurately. The vertex form directly reveals the vertex coordinates, making it a valuable tool for analyzing quadratic functions.
Sandra's attempt to convert the quadratic function to vertex form provides several valuable takeaways:
- Importance of Accurate Calculations: Even a small error in calculation, as seen in Step 3, can lead to an incorrect result. Precision is crucial in mathematical manipulations.
- Understanding the Process: It's not enough to memorize steps; understanding the underlying principles of completing the square is essential for accurate application.
- Vertex Form as a Powerful Tool: Vertex form provides valuable insights into the parabola's vertex and overall shape, making it a powerful tool for analysis.
- Checking Your Work: It's always a good practice to check your work and ensure that each step is logically sound and arithmetically correct.
By dissecting Sandra's work, we gain a deeper understanding of the process of converting quadratic functions to vertex form and the importance of accuracy and conceptual understanding. The ability to convert quadratic functions to vertex form is a valuable skill in algebra and calculus, enabling us to analyze and solve a wide range of problems.
Mathematics. This problem involves algebraic manipulation of quadratic functions, specifically completing the square and converting to vertex form, which are core concepts in mathematics.
