Quadratic Functions Exploring Intercepts And Range For G(x) = Ax^2 + Q

Introduction

In this comprehensive exploration, we delve into the intricacies of quadratic functions, focusing on the function g(x) = ax² + q. This particular form of a quadratic equation provides a foundational understanding of how parabolas behave and how key features like intercepts and range are determined. We will dissect the given information, which includes the x-intercepts S(2, 0) and R, and the y-intercept T(0, 4), to fully characterize the function g(x). Our primary objectives are to accurately determine the range of g and the x-coordinate of point R. This involves a thorough analysis of how the parameters 'a' and 'q' influence the shape and position of the parabola, as well as a step-by-step approach to solving for these values using the given intercepts. By understanding these principles, you'll gain valuable insights into how to manipulate quadratic functions and predict their behavior in various mathematical contexts. This journey will not only enhance your problem-solving skills but also deepen your appreciation for the elegant symmetry and predictability inherent in quadratic relationships. The core of our investigation revolves around deciphering how the intercepts, which serve as crucial landmarks on the graph, dictate the overall structure of the parabolic curve. As we progress, we'll unravel the relationship between these intercepts and the function's parameters, ultimately allowing us to paint a complete picture of the quadratic equation at hand. Understanding the range of a function, in this case, is pivotal as it defines the vertical boundaries within which the function operates. By determining the range, we gain a sense of the function's output possibilities, and this understanding is particularly useful in applied scenarios where we seek to optimize or predict outcomes. The x-coordinate of point R, an x-intercept, is another critical piece of information that helps us anchor the parabola on the coordinate plane. Knowing this value allows us to further refine our understanding of the symmetry and position of the curve, paving the way for more complex analyses and applications of the function.

4.1 Determining the Range of g(x) = ax² + q

The range of a function represents the set of all possible output values (y-values) that the function can produce. For a quadratic function in the form g(x) = ax² + q, the range is significantly influenced by the coefficient 'a' and the constant term 'q'. The coefficient 'a' determines the direction in which the parabola opens: if 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The constant term 'q' represents the vertical shift of the parabola's vertex from the x-axis. The vertex is the point where the parabola changes direction—it's the minimum point if 'a' is positive and the maximum point if 'a' is negative. To accurately determine the range, we must first find the vertex of the parabola. Since we know the x-intercepts are S(2, 0) and R, and parabolas are symmetric, the x-coordinate of the vertex lies exactly midway between these two points. However, we only know one x-intercept, S(2, 0), and need to find the x-coordinate of R. The y-intercept T(0, 4) gives us another crucial piece of information, allowing us to determine the values of 'a' and 'q'. Substituting T(0, 4) into the equation, we get 4 = a(0)² + q, which simplifies to q = 4. This tells us the vertical shift of the parabola is 4 units upwards. The x-intercepts are the points where g(x) = 0. So, we have ax² + 4 = 0 when x = 2. Plugging this in, we get a(2)² + 4 = 0, which simplifies to 4a + 4 = 0. Solving for 'a', we find a = -1. Now we have the complete quadratic function: g(x) = -x² + 4. Since 'a' is negative, the parabola opens downwards, and the vertex is the maximum point. The vertex form of a quadratic equation is g(x) = a(x - h)² + k, where (h, k) is the vertex. In our case, g(x) = -(x - 0)² + 4, so the vertex is (0, 4). Since the parabola opens downwards, the range is all y-values less than or equal to the y-coordinate of the vertex. Therefore, the range of g is y ≤ 4. This range signifies that the function's output values are capped at 4, reflecting the parabolic curve's downward trajectory from its maximum point at the vertex. By understanding these principles, we can confidently navigate the complexities of quadratic functions and their graphical representations. The significance of the vertex cannot be overstated, as it serves as the cornerstone for analyzing the function's behavior and predicting its output values. Moreover, the sign of the leading coefficient 'a' provides immediate insight into the parabola's orientation, allowing us to quickly discern whether the function has a maximum or minimum value. In practical applications, understanding the range of a quadratic function is invaluable for optimizing scenarios, such as maximizing profit or minimizing costs, where quadratic relationships often come into play.

4.2 Finding the x-coordinate of R

Determining the x-coordinate of point R, another x-intercept of the function g(x) = -x² + 4, involves leveraging the properties of parabolas and the information we've already gathered. We know that parabolas are symmetric about their vertex. The vertex, as we found earlier, is at (0, 4). We also know one x-intercept is S(2, 0). The axis of symmetry, which runs vertically through the vertex, acts as a mirror line for the parabola. This means that if there's an x-intercept at x = 2, there must be another x-intercept at an equal distance on the opposite side of the axis of symmetry. The axis of symmetry for our parabola is the y-axis (x = 0), which is the x-coordinate of the vertex. The distance from the x-intercept S(2, 0) to the axis of symmetry is 2 units (from x = 0 to x = 2). Due to the symmetry of the parabola, the other x-intercept R must be the same distance from the axis of symmetry but in the opposite direction. Therefore, the x-coordinate of R is -2. To verify this, we can set g(x) = 0 and solve for x: -x² + 4 = 0. This can be rearranged to x² = 4, and taking the square root of both sides gives us x = ±2. Thus, the x-intercepts are indeed at x = 2 and x = -2. This confirms that the x-coordinate of R is -2. Understanding the symmetry of parabolas is a powerful tool in solving these types of problems. It allows us to quickly deduce information about the function without needing to perform extensive calculations. The concept of symmetry not only simplifies problem-solving but also enhances our understanding of the underlying mathematical principles governing quadratic functions. Furthermore, the ability to visualize the parabola and its symmetry can aid in predicting its behavior and identifying key features such as intercepts and the vertex. In the context of real-world applications, the symmetry of parabolas is evident in various scenarios, from the trajectory of projectiles to the design of parabolic reflectors used in satellite dishes and telescopes. The x-coordinate of R, along with the other intercepts and vertex, provides a comprehensive understanding of the function's behavior and its graphical representation. The ability to accurately determine these features is crucial for a wide range of mathematical and scientific applications. By mastering these concepts, we can confidently tackle more complex problems involving quadratic functions and their properties.

Conclusion

In conclusion, by systematically analyzing the given information and applying the properties of quadratic functions, we have successfully determined the range of g(x) = -x² + 4 to be y ≤ 4 and the x-coordinate of point R to be -2. This exercise underscores the importance of understanding the interplay between the coefficients, intercepts, and symmetry of parabolas. The range of a quadratic function provides crucial information about its output values, while the x-intercepts anchor the parabola on the coordinate plane. The vertex, acting as the parabola's turning point, is central to determining the range and understanding the function's maximum or minimum value. The symmetry inherent in parabolas simplifies problem-solving and allows us to deduce information without complex calculations. This symmetry, manifested around the axis of symmetry, ensures that for every point on one side of the parabola, there's a corresponding point on the other side, equidistant from the axis. This property is particularly useful in finding x-intercepts and understanding the function's behavior. Mastering these concepts equips us with the tools to analyze and predict the behavior of quadratic functions in various mathematical and real-world contexts. From optimizing business processes to understanding physical phenomena, quadratic functions play a significant role in numerous applications. The ability to confidently manipulate and interpret these functions is therefore a valuable skill. The journey through this problem has not only reinforced our understanding of quadratic functions but also highlighted the power of systematic problem-solving and the beauty of mathematical symmetry. By continuing to explore and apply these principles, we can unlock new insights and tackle even more complex mathematical challenges. The foundation laid here will serve as a springboard for further exploration into advanced mathematical concepts and their applications in the world around us. The interplay between theory and practice is essential in mathematics, and by connecting abstract concepts to concrete examples, we deepen our understanding and appreciation for the subject.