Transformations Of Quadratic Functions F(x)=x^2 To P(x)=-50+14x-x^2

When exploring the transformations of functions, particularly in the realm of quadratic equations, it's crucial to grasp how alterations to the original function’s equation manifest graphically. In this article, we will delve into the specific case of transforming the basic quadratic function, f(x) = x^2, to a new quadratic function, p(x) = -50 + 14x - x^2. Our mission is to identify the precise sequence of transformations—shifts, stretches, and reflections—that bridge the gap between these two functions. This involves a methodical approach, emphasizing the importance of completing the square, recognizing vertex form, and meticulously comparing the coefficients and constants within the equations.

The question at hand challenges us to pinpoint which transformation, among a given set of options, accurately describes how the graph of f(x) = x^2 is manipulated to produce the graph of p(x) = -50 + 14x - x^2. The options presented focus on vertical and horizontal shifts, making it essential to understand how these transformations affect the graphical representation of a function. By examining the structure of both equations and applying algebraic techniques, we aim to reveal the underlying transformations. This article will serve as a comprehensive guide, elucidating the step-by-step process involved in solving this problem and providing a robust understanding of quadratic function transformations.

Deconstructing the Functions: f(x) = x^2 and p(x) = -50 + 14x - x^2

To effectively determine the transformations applied to f(x) = x^2 to obtain p(x) = -50 + 14x - x^2, we must first dissect and understand the anatomy of both functions. The function f(x) = x^2 serves as our baseline, the quintessential parabola centered at the origin (0, 0) with its vertex as the minimum point, opening upwards. This simple form allows for straightforward comparisons when identifying transformations. The graph of f(x) = x^2 is symmetrical about the y-axis, a characteristic feature of even functions, and its simplicity makes it an ideal starting point for understanding more complex quadratic functions.

On the other hand, p(x) = -50 + 14x - x^2 presents a slightly more intricate picture. The presence of additional terms, namely the linear term 14x and the constant term -50, along with the negative coefficient of the x^2 term, indicates that several transformations may have been applied. The negative coefficient, in particular, suggests a reflection across the x-axis, which means the parabola will open downwards. The other terms hint at shifts—both horizontal and vertical—that reposition the parabola’s vertex away from the origin.

Furthermore, to fully understand p(x), we need to rewrite it in vertex form, which is expressed as p(x) = a(x - h)^2 + k. This form is immensely useful because it directly reveals the vertex of the parabola at the point (h, k) and the stretch/compression factor a, as well as whether the parabola opens upwards or downwards. By converting p(x) into vertex form, we can directly compare it to f(x) = x^2 and identify the specific transformations applied. This process involves completing the square, a technique that reorganizes the quadratic expression into a perfect square plus a constant, thereby exposing the vertex coordinates and the scaling factor. The subsequent sections will elaborate on this process, providing a clear pathway to identifying the transformations in question.

The Power of Vertex Form: Rewriting p(x) = -50 + 14x - x^2

The vertex form of a quadratic equation, p(x) = a(x - h)^2 + k, is a powerful tool in understanding the transformations applied to a basic quadratic function. It directly reveals the vertex of the parabola at the point (h, k), the vertical stretch or compression factor a, and the direction in which the parabola opens. To transform p(x) = -50 + 14x - x^2 into vertex form, we employ the technique of completing the square. This method allows us to rewrite the quadratic expression as a perfect square plus a constant, thus exposing the key parameters of the parabola.

Firstly, we rearrange the terms of p(x) to group the quadratic and linear terms together: p(x) = -x^2 + 14x - 50. Next, we factor out the coefficient of the x^2 term, which in this case is -1, from the quadratic and linear terms: p(x) = -(x^2 - 14x) - 50. Now, we complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of the x term (which is -14), square it ((-7)^2 = 49), and add it inside the parentheses. However, because we've factored out a -1, we must also subtract -1 * 49 = -49 outside the parentheses to maintain the equation’s balance: p(x) = -(x^2 - 14x + 49) - 50 + 49.

The expression inside the parentheses is now a perfect square trinomial, which can be factored as (x - 7)^2. Thus, our equation becomes p(x) = -(x - 7)^2 - 1. This is the vertex form of the quadratic equation. By comparing this to the standard vertex form p(x) = a(x - h)^2 + k, we can identify the parameters: a = -1, h = 7, and k = -1. These parameters tell us a great deal about the transformations applied to f(x) = x^2 to obtain p(x). The a = -1 indicates a reflection across the x-axis; the h = 7 indicates a horizontal shift of 7 units to the right; and the k = -1 indicates a vertical shift of 1 unit downward. With this information, we can confidently evaluate the given options and pinpoint the correct transformation.

Decoding the Transformations: From f(x) = x^2 to p(x) = -(x - 7)^2 - 1

Having successfully converted p(x) = -50 + 14x - x^2 into vertex form, p(x) = -(x - 7)^2 - 1, we are now well-equipped to decode the transformations that map the graph of f(x) = x^2 onto the graph of p(x). The vertex form provides a clear roadmap, with each parameter revealing a specific transformation. By systematically analyzing these parameters, we can precisely describe how the basic parabola has been manipulated.

The first key parameter is a = -1. This value signifies a reflection across the x-axis. In graphical terms, this means that the parabola, which originally opens upwards for f(x) = x^2, is flipped over the x-axis, causing it to open downwards for p(x). This reflection is a fundamental transformation that alters the concavity of the parabola, turning a minimum vertex into a maximum vertex. The negative sign is the telltale indicator of this reflection, making it a critical component in understanding the overall transformation.

Next, we examine h = 7, which represents a horizontal shift. The term (x - 7) inside the squared expression indicates that the graph has been shifted 7 units to the right. This is because replacing x with (x - 7) effectively moves the entire graph in the positive x-direction. The vertex of the original function, f(x) = x^2, is at (0, 0), and this horizontal shift moves the vertex to (7, k), where k is the vertical position determined by the k value in the vertex form.

Finally, the parameter k = -1 represents a vertical shift. The -1 at the end of the equation shifts the entire graph down by 1 unit. This means that the vertex, which was previously at (7, 0) after the horizontal shift and reflection, is now located at (7, -1). The vertical shift repositions the parabola along the y-axis, completing the transformation of the vertex from the origin to its final position. By considering all these transformations—the reflection across the x-axis, the horizontal shift of 7 units to the right, and the vertical shift of 1 unit downward—we can accurately trace the journey from f(x) = x^2 to p(x) = -(x - 7)^2 - 1.

Identifying the Correct Transformation: A Shift Right by 7 Units

Having deconstructed the transformations required to map f(x) = x^2 onto p(x) = -50 + 14x - x^2, we can now confidently identify the correct transformation from the options provided. Our analysis revealed that p(x) is obtained from f(x) through a reflection across the x-axis, a horizontal shift of 7 units to the right, and a vertical shift of 1 unit downward. Among these transformations, the question specifically asks for one of the applied transformations, and the options focus on shifts.

Option A, “a shift up 7 units,” is incorrect. Our vertex form, p(x) = -(x - 7)^2 - 1, indicates a vertical shift downward, not upward. The -1 term at the end of the equation clearly shows a downward shift of 1 unit, contradicting this option.

Option B, “a shift right 1 unit,” is also incorrect. While there is a horizontal shift involved, it is not of 1 unit. The (x - 7) term in the vertex form indicates a horizontal shift of 7 units to the right, not 1 unit. This discrepancy rules out option B.

Option C, “a shift down 1 unit,” is a valid transformation that we identified. The -1 in the vertex form confirms that the graph is indeed shifted down by 1 unit. However, it's essential to note that this is only one of the transformations applied, and the question asks for one specific transformation that is part of the overall mapping.

Option D, “a shift left 7 units,” is incorrect. The (x - 7) term in the vertex form signifies a shift to the right, not to the left. A shift to the left would be represented by a term of the form (x + 7). Therefore, this option is the opposite of the actual horizontal shift.

Considering all the options and our analysis, we can definitively conclude that the correct transformation is a shift right by 7 units. This corresponds to the (x - 7) term in the vertex form of p(x) and accurately describes one of the key manipulations applied to f(x) = x^2 to obtain p(x) = -50 + 14x - x^2. The process of completing the square and understanding vertex form has been instrumental in arriving at this precise answer.

Conclusion: Mastering Transformations of Quadratic Functions

In conclusion, deciphering the transformations applied to the graph of f(x) = x^2 to produce the graph of p(x) = -50 + 14x - x^2 involved a methodical approach grounded in algebraic manipulation and graphical understanding. By converting p(x) into vertex form, we unveiled the precise shifts, stretches, and reflections that bridge the gap between these two quadratic functions. The journey from the basic parabola to the transformed one highlighted the significance of key parameters within the vertex form equation and their corresponding graphical effects.

The process began with a careful deconstruction of both functions, recognizing f(x) = x^2 as the foundational parabola and p(x) as a transformed variant with potential shifts, reflections, and stretches. The pivotal step of completing the square allowed us to rewrite p(x) in vertex form, p(x) = -(x - 7)^2 - 1, which served as a treasure map to the transformations. This form explicitly revealed a reflection across the x-axis (due to the negative coefficient), a horizontal shift of 7 units to the right, and a vertical shift of 1 unit downward.

By systematically analyzing these transformations, we were able to evaluate the provided options and accurately identify the correct one. The horizontal shift of 7 units to the right emerged as the key transformation explicitly described in the options, aligning perfectly with the (x - 7) term in the vertex form. This exercise underscored the power of vertex form in decoding quadratic function transformations and its utility in graphical analysis.

The ability to transform quadratic functions is a fundamental skill in mathematics, with applications spanning across various fields, including physics, engineering, and computer graphics. Mastering these transformations not only enhances problem-solving capabilities but also provides a deeper appreciation for the elegance and interconnectedness of mathematical concepts. This article has served as a comprehensive guide to navigating these transformations, offering a step-by-step methodology for analyzing quadratic functions and identifying the underlying graphical manipulations. By understanding the interplay between algebraic expressions and their graphical representations, learners can confidently tackle complex transformation problems and cultivate a robust understanding of quadratic functions.