Transforming Quadratic Functions Analyzing F(x) = X² To G(x) = -x² + 6x - 5

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The world of functions in mathematics is fascinating, especially when we delve into the realm of transformations. Transformations allow us to manipulate the graph of a function, shifting it, stretching it, or reflecting it across axes. This article dives deep into a specific case of function transformation, focusing on the parent function f(x) = x² and its transformation into the function g(x) = -x² + 6x - 5. We aim to dissect this transformation, pinpointing the exact shifts and reflections that have occurred. By the end of this exploration, you'll gain a solid understanding of how to analyze quadratic function transformations, a crucial skill in algebra and calculus.

Deconstructing the Parent Function: f(x) = x²

Before we jump into the transformation, let's solidify our understanding of the parent function, f(x) = x². This is the fundamental quadratic function, forming a parabola that opens upwards. Its vertex, the lowest point on the graph, sits perfectly at the origin (0, 0). The axis of symmetry, a vertical line that divides the parabola into two mirror images, is the y-axis (x = 0). Understanding these key features of f(x) = x² is crucial because the transformations we'll explore later build upon this foundation. The simplicity of this function makes it an ideal starting point for understanding more complex transformations. The graph of f(x) = x² is symmetrical, with each y-value having two corresponding x-values (except for the vertex). This symmetry is a characteristic feature of parabolas and plays a significant role in their applications in various fields, including physics and engineering. When visualizing f(x) = x², picture a gentle curve that gradually steepens as it moves away from the vertex. This visual representation will help you intuitively grasp how transformations alter the shape and position of the parabola. Furthermore, understanding the domain and range of f(x) = x² is important. The domain, the set of all possible x-values, is all real numbers. The range, the set of all possible y-values, is all non-negative real numbers (y ≥ 0). These properties are essential for analyzing the impact of transformations on the function's behavior.

Unveiling the Transformed Function: g(x) = -x² + 6x - 5

Now, let's turn our attention to the transformed function, g(x) = -x² + 6x - 5. At first glance, it might seem a bit more complex than our parent function. To truly understand the transformations that have occurred, we need to rewrite this equation in vertex form. Vertex form provides a clear picture of the transformations by explicitly showing the horizontal and vertical shifts, as well as any reflections or stretches. The general form of vertex form is g(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola and 'a' determines the direction and stretch of the parabola. To convert g(x) = -x² + 6x - 5 into vertex form, we'll employ a technique called completing the square. This involves manipulating the quadratic expression to create a perfect square trinomial. The process begins by factoring out the coefficient of the x² term (which is -1 in this case) from the first two terms: g(x) = -(x² - 6x) - 5. Next, we take half of the coefficient of the x term (-6), square it ((-3)² = 9), and add and subtract it inside the parentheses: g(x) = -(x² - 6x + 9 - 9) - 5. Now, we can rewrite the expression inside the parentheses as a perfect square: g(x) = -((x - 3)² - 9) - 5. Distributing the negative sign and simplifying, we get: g(x) = -(x - 3)² + 9 - 5, which finally gives us the vertex form: g(x) = -(x - 3)² + 4. This form reveals a wealth of information about the transformations applied to the parent function.

Deciphering the Transformations: From f(x) to g(x)

With g(x) now in vertex form, g(x) = -(x - 3)² + 4, we can clearly identify the transformations that have been applied to the parent function f(x) = x². Let's break down each transformation step-by-step. First, the negative sign in front of the squared term, -(x - 3)², indicates a reflection across the x-axis. This means the parabola, which originally opened upwards, now opens downwards. Imagine flipping the graph of f(x) = x² over the x-axis; that's the effect of this reflection. Next, the term (x - 3)² represents a horizontal shift. Specifically, it's a shift of 3 units to the right. This is because the vertex of the transformed parabola is at x = 3, compared to the vertex of the parent function at x = 0. Remember that transformations inside the parentheses affect the x-values and operate in the opposite direction of the sign. Finally, the '+ 4' term at the end, -(x - 3)² + 4, indicates a vertical shift of 4 units upwards. This shifts the entire parabola upwards, moving the vertex from the x-axis to y = 4. Combining these transformations, we can visualize the entire process. The parent function f(x) = x² is first reflected across the x-axis, then shifted 3 units to the right, and finally shifted 4 units upwards. This results in the transformed function g(x) = -x² + 6x - 5. Understanding these individual transformations is crucial for predicting the behavior of other transformed functions.

Identifying the Correct Statements: A Synthesis

Now that we've thoroughly analyzed the transformations, let's pinpoint the correct statements that describe the transformation from f(x) = x² to g(x) = -x² + 6x - 5. Based on our analysis, we can confidently state the following:

  1. Reflection across the x-axis: The negative sign in front of the squared term in g(x) clearly indicates a reflection across the x-axis. This flips the parabola upside down, changing its orientation.
  2. Horizontal translation 3 units to the right: The (x - 3)² term in the vertex form signifies a horizontal shift. The '- 3' indicates a shift to the right along the x-axis.
  3. Vertical translation 4 units upward: The '+ 4' term in the vertex form represents a vertical shift. This moves the entire parabola upwards along the y-axis.

These three statements accurately capture the transformations that have occurred. We've systematically deconstructed the functions, identified the individual transformations, and now synthesized our understanding to arrive at these conclusions. This process highlights the power of understanding function transformations in analyzing and manipulating mathematical expressions.

Conclusion: Mastering Function Transformations

In conclusion, the transformation of the parent function f(x) = x² into g(x) = -x² + 6x - 5 is a compelling example of how reflections and translations can alter the graph of a function. By converting g(x) into vertex form, we were able to clearly identify the reflection across the x-axis, the horizontal shift of 3 units to the right, and the vertical shift of 4 units upwards. These transformations collectively reshape and reposition the parabola, demonstrating the versatility of function transformations. Mastering these concepts is crucial for success in algebra, calculus, and beyond. The ability to analyze and predict the behavior of transformed functions is a valuable skill in various fields, from physics and engineering to computer graphics and economics. By practicing and applying these techniques, you can develop a deeper understanding of the mathematical world and its applications. Remember, the key to success lies in understanding the underlying principles and practicing their application to a variety of problems. So, continue exploring, experimenting, and expanding your knowledge of function transformations!