Graphing The Quadratic Function F(x) = (x-1)^2 + 2 A Comprehensive Guide

Understanding quadratic functions is a cornerstone of algebra and calculus. The ability to graph quadratic functions accurately is essential for visualizing their behavior and solving related problems. This article focuses on the quadratic function f(x) = (x-1)^2 + 2, delving into its properties and illustrating how to construct its graph. We will explore the standard form of a quadratic equation, identify key features such as the vertex and axis of symmetry, and discuss how these elements contribute to the overall shape of the parabola. By the end of this discussion, you will have a comprehensive understanding of how to graph quadratic functions like this one and will be able to apply these principles to other quadratic equations.

Understanding the Standard Form

The quadratic function given, f(x) = (x-1)^2 + 2, is presented in vertex form, which is a particularly useful way to represent quadratic equations. The vertex form is generally expressed as f(x) = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction – it’s either the minimum or maximum point of the function. The 'a' value determines the direction the parabola opens (upwards if a > 0, downwards if a < 0) and the vertical stretch or compression of the graph. In our function, f(x) = (x-1)^2 + 2, we can directly identify the vertex by comparing it to the standard vertex form. Here, a = 1, h = 1, and k = 2. This immediately tells us that the vertex of the parabola is at the point (1, 2). Because a = 1, which is positive, the parabola opens upwards, indicating that the vertex represents the minimum point of the function. Understanding the vertex form allows us to quickly grasp the key features of the quadratic function, facilitating the process of graphing it accurately.

Identifying the Vertex and Axis of Symmetry

The vertex is a critical point when graphing quadratic functions. As mentioned earlier, for the function f(x) = (x-1)^2 + 2, the vertex is at (1, 2). This point serves as the turning point of the parabola, where the function transitions from decreasing to increasing. The axis of symmetry is another crucial concept related to the vertex. It is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation for the axis of symmetry is always of the form x = h, where h is the x-coordinate of the vertex. In our case, the axis of symmetry is the vertical line x = 1. This means that the parabola is perfectly mirrored along this line. Knowing the vertex and the axis of symmetry gives us a fundamental framework for sketching the graph. We know the parabola will open upwards from the point (1, 2), and the shape will be symmetrical about the line x = 1. This information significantly simplifies the graphing process, as we can plot the vertex and then use the symmetry to find other points on the parabola. By understanding these core elements, we can begin to construct an accurate representation of the function's graph.

Determining the Shape and Direction of the Parabola

The shape and direction of the parabola are largely determined by the coefficient 'a' in the vertex form of the quadratic equation, f(x) = a(x-h)^2 + k. In our function, f(x) = (x-1)^2 + 2, the value of a is 1. Since a is positive (1 > 0), the parabola opens upwards. This means that the vertex (1, 2) represents the minimum point of the function. If a were negative, the parabola would open downwards, and the vertex would represent the maximum point. The magnitude of a also affects the parabola's width. If |a| > 1, the parabola is narrower (or steeper) compared to the standard parabola f(x) = x^2. If 0 < |a| < 1, the parabola is wider (or flatter). In our case, since a = 1, the parabola has the same width as the standard parabola f(x) = x^2. To further illustrate this, we can plot a few additional points. For example, if we plug in x = 0 into the function, we get f(0) = (0-1)^2 + 2 = 1 + 2 = 3. So, the point (0, 3) lies on the parabola. Due to symmetry, the point (2, 3) will also be on the parabola (since x = 2 is the same distance from the axis of symmetry x = 1 as x = 0). By identifying the direction the parabola opens and understanding its width, we can accurately sketch the curve and visualize the function's behavior.

Plotting Additional Points for Accuracy

While the vertex and axis of symmetry provide a solid foundation for graphing a quadratic function, plotting additional points greatly enhances the accuracy of the graph. To do this, we can choose several x-values and calculate the corresponding f(x) values. It's often helpful to select x-values that are symmetrically positioned around the axis of symmetry x = 1. We already found that when x = 0, f(x) = 3, giving us the point (0, 3). Its symmetrical counterpart across the axis of symmetry is the point (2, 3). Let's consider x = -1. Plugging this into our function, we get f(-1) = (-1-1)^2 + 2 = (-2)^2 + 2 = 4 + 2 = 6. So, the point (-1, 6) lies on the parabola. The symmetrical point to (-1, 6) across the axis of symmetry is (3, 6). Now we have a set of points: (1, 2) (vertex), (0, 3), (2, 3), (-1, 6), and (3, 6). Plotting these points on a coordinate plane allows us to see the overall shape of the parabola more clearly. By connecting these points with a smooth curve, we can create a precise graph of the quadratic function f(x) = (x-1)^2 + 2. Plotting additional points helps ensure that our graph accurately represents the behavior of the function, especially its curvature and spread.

Sketching the Graph

With the vertex, axis of symmetry, and several additional points identified, we are now ready to sketch the graph of the quadratic function f(x) = (x-1)^2 + 2. Start by plotting the vertex (1, 2) on the coordinate plane. This is the lowest point on the parabola since a > 0. Next, draw a dashed vertical line through x = 1 to represent the axis of symmetry. This line serves as a mirror for the parabola. Now, plot the additional points we calculated: (0, 3), (2, 3), (-1, 6), and (3, 6). These points help define the shape and spread of the parabola. Begin sketching the curve by drawing a smooth line that passes through the plotted points. The parabola should open upwards and be symmetrical about the line x = 1. Make sure the curve is smooth and continuous, without any sharp corners. As the parabola extends away from the vertex, it should widen gradually, reflecting the quadratic nature of the function. The graph should extend indefinitely in both upward directions, indicated by arrows at the ends of the curve. The resulting graph provides a visual representation of the function f(x) = (x-1)^2 + 2. It clearly shows the vertex as the minimum point, the symmetry about the line x = 1, and the overall upward-opening parabolic shape. Sketching the graph effectively visualizes the behavior of the function and provides a powerful tool for understanding its properties.

Conclusion

In conclusion, graphing the quadratic function f(x) = (x-1)^2 + 2 involves a systematic approach that leverages the properties of quadratic equations. We began by understanding the vertex form, which allowed us to easily identify the vertex (1, 2) and recognize that the parabola opens upwards due to the positive coefficient a = 1. Determining the axis of symmetry, x = 1, provided a crucial reference for the parabola's symmetrical nature. Plotting additional points, such as (0, 3), (2, 3), (-1, 6), and (3, 6), enhanced the accuracy of the graph, allowing us to sketch the curve with precision. The final graph visually represents the function's behavior, showcasing its minimum point at the vertex and its symmetrical shape. Mastering the techniques for graphing quadratic functions like this one is essential for anyone studying algebra and calculus. It not only reinforces mathematical concepts but also provides a visual tool for solving real-world problems that can be modeled by quadratic equations. The ability to accurately graph quadratic functions opens doors to a deeper understanding of mathematical relationships and their applications.