Graphing Quadratic Function F(x) = (x-3)^2 A Step-by-Step Guide

Jul 12, 2025 by ADMIN 64 views

Sketching the Graph of a Quadratic Function: A Comprehensive Guide to f(x) = (x-3)^2

In this comprehensive guide, we will delve into the intricacies of sketching the graph of the quadratic function f(x) = (x-3)^2. Understanding the properties of quadratic functions is crucial in mathematics, as they appear in various applications, from physics to economics. Our primary focus will be on accurately plotting the graph, identifying the axis of symmetry, and determining the vertex. This step-by-step explanation aims to provide a clear and concise method for both beginners and those seeking a refresher on the topic. By the end of this guide, you will have a solid understanding of how to graph quadratic functions and extract key information from their equations.

Understanding Quadratic Functions

Before we dive into the specifics of f(x) = (x-3)^2, it's essential to understand the general form and properties of quadratic functions. A quadratic function is typically expressed in the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a. When a is positive, the parabola opens upwards, indicating a minimum value, while a negative a results in a downward-opening parabola with a maximum value. The vertex of the parabola is the point where it changes direction—the minimum or maximum point on the curve.

In our case, the function f(x) = (x-3)^2 is a special form of a quadratic function known as the vertex form, which is written as f(x) = a(x - h)^2 + k. Here, a determines the direction and width of the parabola, while the vertex of the parabola is located at the point (h, k). This form is incredibly useful because it directly reveals the vertex, making it easier to sketch the graph. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is given by x = h. Understanding these basic properties is crucial for accurately graphing quadratic functions and interpreting their characteristics.

Now, let's break down the function f(x) = (x-3)^2 further. By comparing it to the vertex form f(x) = a(x - h)^2 + k, we can identify that a = 1, h = 3, and k = 0. Since a is positive (1), the parabola opens upwards. The vertex is at the point (3, 0), and the axis of symmetry is the vertical line x = 3. This preliminary analysis provides a solid foundation for sketching the graph. With the vertex and axis of symmetry known, we can easily plot additional points to create an accurate representation of the parabola. The next sections will guide you through the step-by-step process of graphing this function, ensuring a clear understanding of each element involved.

Step-by-Step Guide to Graphing f(x) = (x-3)^2

To accurately sketch the graph of f(x) = (x-3)^2, we'll follow a step-by-step approach. This will ensure that we capture all the essential features of the parabola, including its vertex, axis of symmetry, and overall shape. Each step is designed to build upon the previous one, providing a clear and systematic method for graphing quadratic functions. By the end of this section, you will have a detailed visual representation of the function and a thorough understanding of its properties.

1. Identify the Vertex

As we discussed earlier, the vertex form of a quadratic function, f(x) = a(x - h)^2 + k, directly gives us the coordinates of the vertex, which are (h, k). For the function f(x) = (x-3)^2, we can see that h = 3 and k = 0. Therefore, the vertex of the parabola is at the point (3, 0). This point is crucial as it represents the minimum value of the function and the turning point of the parabola. Marking the vertex on the coordinate plane is the first key step in sketching the graph accurately.

The vertex acts as the anchor point around which the rest of the parabola is drawn. Its coordinates provide the base from which we can plot additional points, ensuring that the graph is symmetrical and correctly positioned. Understanding how to identify the vertex is fundamental to graphing quadratic functions, and in the case of f(x) = (x-3)^2, it is straightforward thanks to the vertex form of the equation. With the vertex established, we can proceed to determine the axis of symmetry, which further aids in accurately sketching the parabola.

2. Determine the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. In the vertex form f(x) = a(x - h)^2 + k, the equation of the axis of symmetry is given by x = h. For our function, f(x) = (x-3)^2, h is equal to 3. Therefore, the equation for the axis of symmetry is x = 3. This line is crucial for ensuring that the graph of the parabola is symmetric and accurately reflects the function's behavior.

On the coordinate plane, the axis of symmetry is a vertical line that intersects the x-axis at the vertex's x-coordinate, which is 3 in this case. Visualizing the axis of symmetry helps in plotting additional points, as we know that for every point on one side of the axis, there is a corresponding point on the other side at the same vertical distance. This symmetry simplifies the graphing process and ensures a balanced and accurate representation of the parabola. The axis of symmetry not only aids in graphing but also reinforces our understanding of the function's symmetrical nature. With both the vertex and axis of symmetry determined, we can move on to plotting additional points to complete the graph.

3. Plot Additional Points

To accurately sketch the parabola, plotting additional points is essential. Since the parabola is symmetrical, we can choose a few x-values on one side of the axis of symmetry and calculate the corresponding y-values. These points, along with their symmetric counterparts on the other side of the axis, will give us a clear shape of the curve. For the function f(x) = (x-3)^2, we've already identified the vertex at (3, 0) and the axis of symmetry as x = 3. Now, let’s choose some x-values greater than 3 and calculate the corresponding f(x) values.

Let's start with x = 4. Plugging this into the function, we get f(4) = (4-3)^2 = 1^2 = 1. So, the point (4, 1) is on the graph. Due to symmetry, there's a corresponding point on the other side of the axis of symmetry at x = 2, which will also have f(2) = (2-3)^2 = (-1)^2 = 1. This gives us the point (2, 1). Next, let’s try x = 5. We calculate f(5) = (5-3)^2 = 2^2 = 4. This gives us the point (5, 4), and its symmetrical counterpart at x = 1, where f(1) = (1-3)^2 = (-2)^2 = 4, resulting in the point (1, 4). By plotting these points and connecting them with a smooth curve, we can accurately sketch the parabola.

The more points we plot, the more accurate our graph will be. However, a few well-chosen points around the vertex are usually sufficient to get a good representation. These points help to define the parabola's width and curvature, ensuring that the graph is a true reflection of the function f(x) = (x-3)^2. With these additional points plotted, we have a comprehensive view of the parabola's shape and position on the coordinate plane.

4. Sketch the Graph

With the vertex at (3, 0), the axis of symmetry at x = 3, and additional points such as (2, 1), (4, 1), (1, 4), and (5, 4) plotted, we can now sketch the graph of the quadratic function f(x) = (x-3)^2. The parabola opens upwards because the coefficient a in front of the squared term is positive (in this case, a = 1). Starting from the vertex, draw a smooth, U-shaped curve that passes through the plotted points, ensuring that the curve is symmetrical about the axis of symmetry.

The symmetry of the parabola is crucial to maintain while sketching. The distances of the points from the axis of symmetry should be equal on both sides. This ensures that the graph accurately represents the function. As you sketch the curve, pay attention to the curvature and ensure that it widens smoothly as you move away from the vertex. The vertex represents the minimum point of the function, and the parabola extends upwards from this point.

To complete the graph, extend the curve beyond the plotted points, indicating that the parabola continues indefinitely in both directions. This final step gives a complete visual representation of the function f(x) = (x-3)^2. By following these steps—identifying the vertex, determining the axis of symmetry, plotting additional points, and sketching the curve—you can accurately graph any quadratic function given in vertex form. This process not only provides a visual understanding of the function but also reinforces the relationship between the algebraic form and its graphical representation.

Identifying Key Features of the Graph

Once the graph of the quadratic function f(x) = (x-3)^2 is sketched, it's important to identify and state its key features. These features provide a concise summary of the function's behavior and properties. The main features we'll focus on are the vertex, the axis of symmetry, and the overall shape of the parabola. These elements are fundamental in understanding the nature and characteristics of quadratic functions.

Vertex

The vertex is the turning point of the parabola and, for f(x) = (x-3)^2, it is the minimum point since the parabola opens upwards. As we determined earlier, the vertex is located at the point (3, 0). This means that the minimum value of the function occurs when x = 3, and the value of the function at this point is f(3) = 0. The vertex is a critical feature because it gives us the lowest (or highest, for downward-opening parabolas) point on the graph and serves as a reference for understanding the function's range.

The coordinates of the vertex are directly derived from the vertex form of the equation, f(x) = a(x - h)^2 + k, where (h, k) represents the vertex. In our case, (h, k) is (3, 0), making it straightforward to identify the vertex. Recognizing the vertex is not only essential for graphing but also for solving optimization problems, where we need to find the minimum or maximum value of a quadratic function. The vertex provides this crucial information, making it a key feature to identify and state when analyzing quadratic functions.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For the function f(x) = (x-3)^2, the equation of the axis of symmetry is x = 3. This line passes through the vertex, and every point on the parabola has a corresponding point on the other side of this line at the same vertical distance. The axis of symmetry is an essential feature because it highlights the symmetrical nature of quadratic functions and simplifies the process of graphing.

The equation of the axis of symmetry is directly related to the x-coordinate of the vertex, which, in this case, is 3. Therefore, the vertical line x = 3 represents the axis of symmetry. Understanding the axis of symmetry helps in plotting points, as for every point (x, y) on the parabola, there is a corresponding point (2h - x, y), where h is the x-coordinate of the vertex. This symmetry simplifies the process of sketching the graph and ensures its accuracy. Stating the equation of the axis of symmetry is crucial for a complete description of the quadratic function's graphical properties.

Shape of the Parabola

The overall shape of the parabola is determined by the coefficient a in the quadratic function f(x) = a(x - h)^2 + k. In the case of f(x) = (x-3)^2, a = 1, which is positive. This indicates that the parabola opens upwards, meaning it has a minimum value at the vertex. The value of a also affects the width of the parabola; a larger absolute value of a results in a narrower parabola, while a smaller absolute value makes it wider.

Since a = 1, the parabola has a standard width, neither too narrow nor too wide. The upward-opening nature of the parabola is a direct consequence of the positive coefficient a, and this is a key characteristic to note when describing the shape. The shape of the parabola, along with the vertex and axis of symmetry, provides a complete picture of the quadratic function's graphical behavior. By identifying and stating these features, we can effectively communicate the essential properties of the function and its representation on the coordinate plane.

Conclusion

In conclusion, sketching the graph of the quadratic function f(x) = (x-3)^2 involves a systematic approach that includes identifying the vertex, determining the axis of symmetry, plotting additional points, and finally, sketching the curve. The vertex, located at (3, 0), represents the minimum point of the function, while the axis of symmetry, given by the equation x = 3, divides the parabola into two symmetrical halves. The positive coefficient a (which is 1 in this case) indicates that the parabola opens upwards, giving it a U-shape. By plotting additional points and maintaining symmetry about the axis, we can accurately sketch the graph, providing a visual representation of the function's behavior.

Understanding these key features is crucial for analyzing quadratic functions and their applications. The vertex gives us the minimum or maximum value of the function, the axis of symmetry highlights its symmetrical nature, and the shape tells us how the function behaves as x varies. This comprehensive approach not only helps in graphing but also in solving related problems, such as finding the range of the function or determining its intercepts. By mastering these techniques, you can confidently sketch and analyze quadratic functions, laying a strong foundation for further mathematical studies.

This guide has provided a detailed explanation of how to sketch the graph of f(x) = (x-3)^2 and identify its key features. By following these steps, you can accurately represent any quadratic function in vertex form, enhancing your understanding of mathematical concepts and problem-solving skills.