Transformations Of Quadratic Functions F(x) = X^2 To H(x) = (1/5)x^2 + 12

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Introduction

In the realm of mathematics, understanding how transformations affect functions is crucial for grasping the behavior and properties of various graphs. Among these functions, quadratic functions hold a significant place due to their wide applicability in modeling real-world phenomena. This article delves into the specific transformation of a basic quadratic function, f(x) = x², into a modified function, h(x) = (1/5)x² + 12. Our focus will be on analyzing the effects of these transformations on the graph of the original function, providing a comprehensive understanding of vertical compression and vertical shifts. We will explore the underlying principles, visual representations, and mathematical justifications for these transformations, ensuring a clear and thorough grasp of the concepts involved.

The Parent Function: f(x) = x²

Before we delve into the transformations, it is essential to understand the parent function, f(x) = x². This is the most basic quadratic function, and its graph is a parabola that opens upwards, with its vertex at the origin (0, 0). The shape of this parabola serves as the foundation for understanding how transformations alter the graph. The key characteristics of f(x) = x² include its symmetry about the y-axis, its minimum value at the vertex, and its increasing rate of change as x moves away from the vertex in either direction. This basic understanding is vital for comparing and contrasting the transformed function h(x) and discerning the impact of the applied transformations. The graph of f(x) = x² serves as a baseline, allowing us to visually and mathematically assess how the alterations in the function's equation manifest as changes in its graphical representation.

Key Features of f(x) = x²

To fully appreciate the transformations, let's highlight some key features of f(x) = x²:

  • Vertex: The vertex is at the point (0, 0), representing the minimum value of the function.
  • Symmetry: The parabola is symmetric about the y-axis, meaning that for any x, f(x) = f(-x).
  • Direction: The parabola opens upwards, indicating a positive leading coefficient (the coefficient of the x² term).
  • Rate of Change: The function increases more rapidly as |x| increases. This means the parabola becomes steeper as we move away from the vertex along the x-axis.

These characteristics provide a framework for analyzing how the transformations in h(x) modify the graph's shape and position. By comparing these features with those of the transformed function, we can accurately describe the effects of the transformations.

Vertical Compression: The Role of the 1/5 Factor

The first transformation we encounter in h(x) = (1/5)x² + 12 is the multiplication of the x² term by the fraction 1/5. This factor plays a crucial role in vertically compressing the graph. Vertical compression occurs when the y-values of the function are scaled down, effectively making the graph appear wider or flatter compared to the original function. In this case, multiplying x² by 1/5 means that every y-value of f(x) = x² is reduced to one-fifth of its original value. This compression significantly impacts the parabola's shape, causing it to stretch out along the x-axis. The visual effect is a flattening of the parabola, making it less steep than the parent function. The vertex remains on the y-axis, but the overall appearance of the graph is noticeably altered due to this vertical compression.

Mathematical Explanation of Vertical Compression

Mathematically, the transformation g(x) = af(x)* results in vertical compression if 0 < |a| < 1. In our case, a = 1/5, which clearly falls within this range. This means that the y-coordinate of every point on the graph of f(x) is multiplied by 1/5 to obtain the corresponding point on the graph of g(x). For example, consider the point (2, 4) on f(x) = x². When vertically compressed by a factor of 1/5, the new point becomes (2, 4/5). This scaling down of the y-values is what creates the visual effect of vertical compression. The compression factor (1/5 in this case) directly influences how much the graph is flattened; a smaller fraction leads to a more significant compression.

Visualizing Vertical Compression

To visualize vertical compression, imagine pressing down on the parabola from above. This action would flatten the curve, making it wider. The vertical compression by a factor of 1/5 has a similar effect on the graph of f(x) = x². The parabola appears stretched horizontally, as the y-values are closer to the x-axis. This visual representation helps to solidify the understanding of how the factor 1/5 transforms the shape of the graph. By comparing the graphs of f(x) = x² and (1/5)x², the vertical compression becomes readily apparent.

Vertical Shift: The Impact of the +12 Constant

The second transformation in h(x) = (1/5)x² + 12 is the addition of the constant 12. This constant term induces a vertical shift of the graph. Specifically, adding 12 to the function shifts the entire graph upwards by 12 units. This means that every point on the graph of (1/5)x² is moved vertically upwards by 12 units, resulting in a parallel translation of the parabola. The shape of the parabola remains unchanged, but its position on the coordinate plane is altered. The vertex, which was previously at (0, 0) for f(x) = x² and remained at (0, 0) after the vertical compression, is now shifted upwards to (0, 12) in the graph of h(x). This vertical shift is a fundamental transformation that affects the y-intercept and the overall vertical positioning of the function's graph.

Mathematical Explanation of Vertical Shift

Mathematically, the transformation g(x) = f(x) + c results in a vertical shift. If c is positive, the graph shifts upwards by c units; if c is negative, the graph shifts downwards by |c| units. In our case, c = 12, which is positive. Therefore, the graph of (1/5)x² is shifted upwards by 12 units to obtain the graph of h(x) = (1/5)x² + 12. This means that the y-coordinate of every point on the graph of (1/5)x² is increased by 12 to obtain the corresponding point on the graph of h(x). For example, the vertex (0, 0) on the compressed graph becomes (0, 12) after the vertical shift. This constant addition directly influences the vertical position of the graph without altering its shape or orientation.

Visualizing Vertical Shift

To visualize a vertical shift, imagine picking up the entire graph and moving it upwards (or downwards, depending on the sign of c). In this case, we are lifting the graph of (1/5)x² by 12 units. This movement does not change the parabola's width or curvature; it simply repositions it on the coordinate plane. The vertical shift is readily apparent when comparing the graphs of (1/5)x² and h(x) = (1/5)x² + 12. The entire parabola is translated upwards, making the y-intercept of h(x) equal to 12. This visual understanding helps to clarify the impact of the constant term on the graph's position.

Combining Transformations: From f(x) = x² to h(x) = (1/5)x² + 12

Now, let's consolidate our understanding by examining the combined effect of the vertical compression and vertical shift on the graph of f(x) = x². The transformation from f(x) = x² to h(x) = (1/5)x² + 12 involves two sequential steps:

  1. Vertical Compression: The graph of f(x) = x² is vertically compressed by a factor of 1/5, resulting in a wider, flatter parabola represented by the equation g(x) = (1/5)x².
  2. Vertical Shift: The compressed graph is then shifted upwards by 12 units, leading to the final transformed function h(x) = (1/5)x² + 12.

These two transformations work in tandem to alter the original parabola's shape and position. The vertical compression flattens the parabola, while the vertical shift repositions it higher on the coordinate plane. Understanding the order in which these transformations are applied is crucial. The vertical compression scales the y-values, and then the vertical shift adds a constant to these scaled values. This sequential application of transformations allows us to accurately predict and interpret the final graph of h(x).

Summary of the Transformations

In summary, the graph of f(x) = x² undergoes the following transformations to become h(x) = (1/5)x² + 12:

  • Vertical Compression by a Factor of 1/5: This makes the parabola wider and flatter.
  • Vertical Shift Upwards by 12 Units: This moves the entire graph upwards, changing the vertex from (0, 0) to (0, 12).

Conclusion

In conclusion, the transformation of f(x) = x² to h(x) = (1/5)x² + 12 demonstrates the fundamental principles of vertical compression and vertical shifts in function transformations. The factor of 1/5 vertically compresses the graph, making it wider, while the constant +12 shifts the graph upwards by 12 units. By understanding these transformations, we gain a deeper insight into how changes in a function's equation directly impact its graphical representation. This knowledge is essential for analyzing and interpreting various mathematical models and real-world applications involving quadratic functions. Mastering these concepts allows for a more intuitive and comprehensive understanding of function behavior and graphical transformations in mathematics.

This detailed analysis not only clarifies the specific transformations but also provides a framework for understanding more complex transformations of functions. The principles discussed here can be applied to various other types of functions, making this knowledge invaluable for students and professionals alike. The ability to deconstruct and interpret transformations is a cornerstone of mathematical literacy, enabling us to effectively model and analyze the world around us.