Transformations Of Quadratic Functions Finding G(x) After Shifts

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In the realm of mathematics, transformations play a pivotal role in altering the characteristics of functions. Among the various types of functions, quadratic functions, defined by the general form f(x) = ax^2 + bx + c, hold a prominent position. These functions are characterized by their parabolic shape, which can be manipulated through transformations such as shifts, stretches, and reflections. In this comprehensive exploration, we will delve into the intricacies of transformations applied to quadratic functions, specifically focusing on vertical and horizontal shifts. We will examine how these transformations affect the function's equation and graph, ultimately leading us to determine the new function g(x) resulting from the specified transformations. Our focus is on the given function f(x) = x^2 and the transformations applied to it: a vertical shift upward by 44 units and a horizontal shift to the right by 97 units. By understanding the principles behind these transformations, we can effectively manipulate quadratic functions to achieve desired outcomes in various mathematical and real-world applications. We will break down the process step-by-step, ensuring a clear and thorough understanding of the concepts involved. This detailed analysis will not only provide the solution to the specific problem but also equip readers with the knowledge to tackle similar problems involving transformations of quadratic functions.

Vertical Shifts: Translating the Parabola Upward

Vertical shifts involve moving the graph of a function up or down along the y-axis. In the context of quadratic functions, a vertical shift alters the vertical position of the parabola's vertex, which is the point where the parabola changes direction. The general rule for a vertical shift is that if we add a constant k to the function, the graph shifts upward by k units. Conversely, if we subtract k from the function, the graph shifts downward by k units. This can be represented mathematically as follows:

  • If g(x) = f(x) + k, the graph of f(x) is shifted upward by k units.
  • If g(x) = f(x) - k, the graph of f(x) is shifted downward by k units.

In our specific case, we are given the function f(x) = x^2 and a vertical shift upward by 44 units. Applying the rule for vertical shifts, we add 44 to the function f(x). This results in a new function, which we can denote as h(x), representing the function after the vertical shift:

  • h(x) = f(x) + 44 = x^2 + 44

The function h(x) = x^2 + 44 represents the original parabola f(x) = x^2 shifted upward by 44 units. The vertex of the original parabola f(x) = x^2 is at the origin (0, 0). After the vertical shift, the vertex of the new parabola h(x) = x^2 + 44 is at the point (0, 44). This vertical shift does not affect the shape or width of the parabola; it simply moves it vertically along the y-axis. Understanding vertical shifts is crucial for manipulating quadratic functions to fit specific conditions or models, allowing us to adjust the function's vertical position without altering its fundamental parabolic nature. This is a key concept in various applications, including physics, engineering, and economics, where quadratic functions are used to model various phenomena. The simplicity of vertical shifts makes them a fundamental building block in the broader topic of function transformations.

Horizontal Shifts: Translating the Parabola Left or Right

Horizontal shifts involve moving the graph of a function left or right along the x-axis. Unlike vertical shifts, horizontal shifts affect the function's argument, which is the input to the function. The general rule for a horizontal shift is that if we replace x with (x - h) in the function, the graph shifts to the right by h units. Conversely, if we replace x with (x + h), the graph shifts to the left by h units. This can be represented mathematically as follows:

  • If g(x) = f(x - h), the graph of f(x) is shifted to the right by h units.
  • If g(x) = f(x + h), the graph of f(x) is shifted to the left by h units.

It's important to note the counterintuitive nature of horizontal shifts: a subtraction within the argument results in a shift to the right, and an addition results in a shift to the left. This is because the transformation affects the x-values required to produce the same y-values as the original function. In our specific case, after the vertical shift, we have the function h(x) = x^2 + 44. Now, we need to apply a horizontal shift of 97 units to the right. According to the rule for horizontal shifts, we replace x with (x - 97) in the function h(x). This gives us the new function g(x):

  • g(x) = h(x - 97) = (x - 97)^2 + 44

The function g(x) = (x - 97)^2 + 44 represents the parabola h(x) = x^2 + 44 shifted 97 units to the right. The vertex of h(x) was at (0, 44). After the horizontal shift, the vertex of the new parabola g(x) is at the point (97, 44). This horizontal shift moves the parabola along the x-axis without altering its shape or vertical position relative to the y-axis. Understanding horizontal shifts is crucial for positioning quadratic functions accurately in models and applications, particularly when the location of the vertex is a critical parameter. By combining horizontal and vertical shifts, we can precisely control the position of a parabola in the coordinate plane, allowing us to tailor the function to specific requirements.

Combining Vertical and Horizontal Shifts: Finding g(x)

In the previous sections, we explored vertical and horizontal shifts separately. Now, let's combine these transformations to find the final function g(x) after applying both shifts to the original function f(x) = x^2. We first performed a vertical shift upward by 44 units, which resulted in the function h(x) = x^2 + 44. Then, we performed a horizontal shift to the right by 97 units, which involved replacing x with (x - 97) in the function h(x). This led us to the function g(x) = (x - 97)^2 + 44. The function g(x) represents the original parabola f(x) = x^2 after both transformations have been applied. The vertex of the original parabola was at (0, 0). After the vertical shift, the vertex moved to (0, 44). Finally, after the horizontal shift, the vertex moved to (97, 44). This combined transformation demonstrates how we can manipulate quadratic functions by independently controlling their vertical and horizontal positions. The order in which we apply these shifts does not affect the final result, as the transformations are independent of each other. We could have performed the horizontal shift first and then the vertical shift, and we would have arrived at the same function g(x). This flexibility is a valuable aspect of function transformations, allowing us to approach problems from different perspectives and simplify the process as needed. The ability to combine transformations is a powerful tool in mathematical modeling, enabling us to create complex functions by building upon simpler ones. By understanding how vertical and horizontal shifts interact, we can effectively manipulate quadratic functions to represent a wide range of phenomena.

Determining the New Function g(x) After Transformations

Having explored the individual and combined effects of vertical and horizontal shifts, we can now definitively determine the new function g(x) resulting from the specified transformations applied to f(x) = x^2. The transformations involved a vertical shift upward by 44 units and a horizontal shift to the right by 97 units. We systematically applied these transformations, first shifting the function vertically to obtain h(x) = x^2 + 44, and then shifting it horizontally to obtain g(x) = (x - 97)^2 + 44. Therefore, the new function g(x) is given by:

  • g(x) = (x - 97)^2 + 44

This function represents a parabola with its vertex at the point (97, 44). The parabola has the same shape and width as the original function f(x) = x^2, but it has been translated 97 units to the right and 44 units upward. The equation g(x) = (x - 97)^2 + 44 is in vertex form, which is a useful representation for quadratic functions as it directly reveals the vertex coordinates. The vertex form of a quadratic function is given by g(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In our case, a = 1, h = 97, and k = 44, confirming that the vertex is indeed at (97, 44). The function g(x) can also be expanded to the standard form g(x) = ax^2 + bx + c, but the vertex form provides a more intuitive understanding of the function's transformations. The process of determining g(x) highlights the power of function transformations in manipulating graphs and equations. By applying vertical and horizontal shifts, we can precisely position a parabola in the coordinate plane, making it a valuable tool in various mathematical and practical applications.

Conclusion

In conclusion, we have successfully determined the new function g(x) after applying a vertical shift upward by 44 units and a horizontal shift to the right by 97 units to the original function f(x) = x^2. Through a step-by-step analysis, we first considered the vertical shift, which resulted in the intermediate function h(x) = x^2 + 44. Subsequently, we applied the horizontal shift, replacing x with (x - 97), which led us to the final function g(x) = (x - 97)^2 + 44. This function represents the parabola f(x) = x^2 translated 97 units to the right and 44 units upward, with its vertex located at (97, 44). The process of finding g(x) demonstrates the fundamental principles of function transformations, particularly vertical and horizontal shifts. These transformations are essential tools in manipulating functions to achieve desired positions and orientations in the coordinate plane. The ability to combine transformations allows us to create complex functions from simpler ones, making it a crucial concept in mathematical modeling and various applications. Understanding these transformations provides a solid foundation for further exploration of function behavior and manipulation. The systematic approach we employed, breaking down the problem into individual transformations, is a valuable strategy for tackling similar problems in mathematics. The final function, g(x) = (x - 97)^2 + 44, serves as a clear and concise representation of the transformations applied to the original function, highlighting the power and versatility of mathematical transformations in shaping functions and their graphs.