Finding The Vertex Of A Quadratic Function A Step-by-Step Guide
Charla is on a mission! She's determined to find the vertex of the quadratic function f(x) = x^2 - 18x + 60 by transforming it into the coveted vertex form. This exploration will not only pinpoint the vertex but also illuminate the function's key characteristics. Let's dive into the world of quadratic functions and discover how to conquer this challenge.
Understanding Quadratic Functions and Vertex Form
Quadratic functions, the stars of our show, are polynomial functions of degree two. They gracefully curve into a U-shape, known as a parabola. The standard form of a quadratic function, f(x) = ax^2 + bx + c, reveals the coefficients a, b, and c, each playing a crucial role in shaping the parabola. The vertex form, on the other hand, f(x) = a(x - h)^2 + k, is the key to directly unlocking the vertex coordinates (h, k). The vertex, that pivotal point, represents either the minimum (if a > 0) or the maximum (if a < 0) value of the function. Understanding the vertex is crucial as it reveals the function's extreme point and axis of symmetry, providing a comprehensive view of the parabola's behavior. The x-coordinate of the vertex, h, also defines the axis of symmetry, a vertical line that perfectly mirrors the parabola. The y-coordinate, k, reveals the minimum or maximum value of the function, depending on the parabola's orientation (whether it opens upwards or downwards). So, by transforming the function into vertex form, we gain direct access to these critical features, making analysis and graphing much simpler. The coefficient a in both forms plays a significant role; it dictates the direction the parabola opens (upwards if a is positive, downwards if a is negative) and influences its width (a larger absolute value of a results in a narrower parabola). Thus, mastering the vertex form is essential for anyone seeking to understand and manipulate quadratic functions effectively. This form not only simplifies the identification of key characteristics but also provides a solid foundation for solving quadratic equations and inequalities.
Charla's Strategy Completing the Square
To convert the given quadratic function, f(x) = x^2 - 18x + 60, into vertex form, Charla employs the powerful technique of completing the square. Completing the square is an algebraic maneuver that transforms a quadratic expression into a perfect square trinomial, making it easily factorable and revealing the vertex form. The initial steps involve focusing on the x^2 and x terms. In our case, these terms are x^2 and -18x. The goal is to manipulate these terms to create a perfect square. This is achieved by taking half of the coefficient of the x term (-18), squaring it, and adding it to the expression. Half of -18 is -9, and squaring -9 gives us 81. However, we can't just add 81 without compensating for it, so we'll add and subtract 81 within the equation. This ensures that we're not changing the value of the function, only its form. The expression now looks like x^2 - 18x + 81 - 81 + 60. The first three terms, x^2 - 18x + 81, now form a perfect square trinomial, which can be factored as (x - 9)^2. Combining the constants, -81 and +60, gives us -21. Thus, the function in vertex form becomes f(x) = (x - 9)^2 - 21. This transformation unveils the vertex coordinates directly. The vertex is located at the point (9, -21), where 9 is the x-coordinate and -21 is the y-coordinate. Completing the square is a versatile technique applicable not only to finding the vertex but also to solving quadratic equations and simplifying expressions. It's a fundamental skill in algebra and a cornerstone for understanding quadratic functions.
Unveiling the Vertex (9, -21)
After the successful completion of the square, Charla has transformed the function into its vertex form: f(x) = (x - 9)^2 - 21. This form immediately reveals the vertex of the parabola. By comparing this to the general vertex form, f(x) = a(x - h)^2 + k, we can directly identify the vertex coordinates as (h, k) = (9, -21). This means the vertex is located at the point where x equals 9 and y equals -21 on the coordinate plane. The x-coordinate, 9, indicates the axis of symmetry for the parabola, a vertical line that divides the parabola into two symmetrical halves. The y-coordinate, -21, is the minimum value of the function since the coefficient of the (x - 9)^2 term is positive (which means the parabola opens upwards). Therefore, the parabola's lowest point is at y = -21. Understanding the significance of the vertex is paramount in analyzing quadratic functions. It provides a crucial reference point for sketching the graph, determining the range of the function, and solving optimization problems. In this case, knowing that the vertex is at (9, -21) allows us to quickly visualize the parabola's shape and position. We know it opens upwards, its axis of symmetry is the vertical line x = 9, and its minimum value is -21. These insights are invaluable for various applications, from modeling projectile motion to designing parabolic mirrors.
Analyzing the Vertex's y-coordinate
The crucial question now is: What does the y-coordinate of the vertex tell us about the function? We've established that the vertex is (9, -21). The y-coordinate, -21, is the key to understanding the function's minimum value. Because the coefficient of the x^2 term in the original function f(x) = x^2 - 18x + 60 is positive (it's 1), the parabola opens upwards. This upward-opening shape means the vertex represents the lowest point on the graph. Therefore, the y-coordinate of the vertex, -21, is the minimum value of the function. This tells us that the function f(x) will never produce a value lower than -21. For any value of x we input into the function, the output f(x) will always be greater than or equal to -21. This understanding is essential for various applications. For example, if this function modeled the cost of production, we would know that the minimum cost is -21 units. If it modeled the height of a projectile, we would know that the minimum height (or ground level) is -21 units (in this context, the negative value might represent a point below a reference level). In the context of Charla's challenge, we can definitively say that the y-coordinate of the vertex, -21, is less than zero. This contrasts with any statement suggesting the y-coordinate is greater than zero. This highlights the importance of not just finding the vertex but also interpreting its meaning within the context of the problem.
The Correct Statement
Based on our analysis, the correct statement about the vertex of the function f(x) = x^2 - 18x + 60 is that the y-coordinate of the vertex is less than zero. We found the vertex to be (9, -21), and the y-coordinate, -21, is indeed a negative value. This conclusion stems directly from the process of completing the square and transforming the function into vertex form. The vertex form, f(x) = (x - 9)^2 - 21, clearly reveals the vertex coordinates and makes it easy to determine the sign of the y-coordinate. This problem exemplifies the power of algebraic manipulation in uncovering key characteristics of functions. By completing the square, Charla didn't just find the vertex; she also gained a deeper understanding of the function's behavior. Knowing the vertex's location and the direction the parabola opens allows us to sketch the graph, determine the range, and solve related problems. In this specific case, the negative y-coordinate indicates that the minimum value of the function is negative, which might have significant implications depending on the real-world context the function represents. The ability to accurately determine and interpret the vertex of a quadratic function is a fundamental skill in mathematics, with applications spanning various fields, from physics and engineering to economics and computer science.
Conclusion
Charla's journey to find the vertex of f(x) = x^2 - 18x + 60 showcases the elegance and utility of completing the square. By transforming the function into vertex form, she not only pinpointed the vertex at (9, -21) but also illuminated the function's minimum value and axis of symmetry. The key takeaway is that the y-coordinate of the vertex, -21, is less than zero, a crucial piece of information for understanding the function's behavior. This exploration underscores the importance of mastering algebraic techniques and their applications in analyzing functions. The vertex form serves as a powerful tool, unlocking the secrets hidden within quadratic equations and revealing the parabola's essential features. Charla's success highlights the problem-solving process in mathematics: transforming a problem into a manageable form, applying appropriate techniques, and interpreting the results in a meaningful way. The ability to find and interpret the vertex is a valuable asset, not just in mathematical contexts but also in real-world applications where quadratic functions model diverse phenomena. From optimizing processes to predicting trajectories, the vertex plays a vital role in understanding and manipulating the world around us. Therefore, mastering the art of finding the vertex is a significant step towards mathematical proficiency and problem-solving prowess.
