Graphing F(x) = (x+1)(x-5) A Step-by-Step Guide

In the realm of mathematics, understanding how to graph functions is a fundamental skill. Among various types of functions, quadratic functions hold a significant place due to their wide applications in diverse fields such as physics, engineering, and economics. This article provides a detailed, step-by-step guide on graphing the specific quadratic function f(x) = (x+1)(x-5). We will explore how to identify key features of the graph, including the x-intercepts, the axis of symmetry, and the vertex, which will allow us to accurately sketch the parabola represented by this function. This comprehensive approach aims to enhance your understanding and proficiency in graphing quadratic functions, equipping you with the necessary tools to tackle more complex mathematical problems. By the end of this guide, you will be able to confidently graph similar quadratic functions and interpret their graphical representations.

The x-intercepts of a function are the points where the graph intersects the x-axis. These points are crucial for understanding the behavior of the function and sketching its graph. To find the x-intercepts of the quadratic function f(x) = (x+1)(x-5), we need to determine the values of x for which f(x) = 0. This is because any point on the x-axis has a y-coordinate of 0. Setting f(x) = 0 gives us the equation (x+1)(x-5) = 0. This equation is already factored, which simplifies the process of finding the solutions. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we have two possible cases: x + 1 = 0 or x - 5 = 0. Solving the first equation, x + 1 = 0, we subtract 1 from both sides to get x = -1. This means that the function intersects the x-axis at the point (-1, 0). Solving the second equation, x - 5 = 0, we add 5 to both sides to get x = 5. This means that the function intersects the x-axis at the point (5, 0). Thus, the x-intercepts of the function f(x) = (x+1)(x-5) are (-1, 0) and (5, 0). These points provide the foundation for sketching the parabola, as they define where the graph crosses the horizontal axis. Understanding how to find x-intercepts is essential for analyzing quadratic functions and their graphs. By identifying these key points, we gain valuable insights into the function's behavior and its position in the coordinate plane.

The midpoint between the x-intercepts plays a crucial role in graphing quadratic functions, as it helps us determine the axis of symmetry and the vertex of the parabola. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves, and the vertex is the point where the parabola reaches its minimum or maximum value. To find the midpoint between the x-intercepts (-1, 0) and (5, 0), we use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are given by (((x1 + x2) / 2), ((y1 + y2) / 2)). In this case, our x-intercepts are (-1, 0) and (5, 0), so we have x1 = -1, y1 = 0, x2 = 5, and y2 = 0. Plugging these values into the midpoint formula, we get the x-coordinate of the midpoint as ((-1 + 5) / 2) = (4 / 2) = 2. The y-coordinate of the midpoint is ((0 + 0) / 2) = 0. Therefore, the midpoint between the x-intercepts is (2, 0). This point lies on the axis of symmetry, which is a vertical line passing through x = 2. The axis of symmetry is a fundamental characteristic of parabolas, as it reflects the symmetry of the graph. The vertex of the parabola will also lie on this line, making the midpoint calculation a crucial step in determining the vertex's location. By finding the midpoint, we have identified the x-coordinate of the vertex, which is essential for completing the graph of the quadratic function. Understanding the relationship between the x-intercepts, the midpoint, the axis of symmetry, and the vertex is key to accurately graphing quadratic functions.

The vertex of a parabola is the point where the function reaches its minimum or maximum value, and it is a critical feature for graphing quadratic functions. For a parabola that opens upwards (like the one in our example), the vertex represents the minimum point. We already found the x-coordinate of the vertex in the previous step by calculating the midpoint between the x-intercepts, which is x = 2. To find the y-coordinate of the vertex, we need to evaluate the function f(x) = (x+1)(x-5) at x = 2. Substituting x = 2 into the function, we get f(2) = (2+1)(2-5) = (3)(-3) = -9. Therefore, the vertex of the parabola is the point (2, -9). This point represents the lowest point on the graph of the function. The vertex, along with the x-intercepts, provides a clear picture of the parabola's shape and position in the coordinate plane. Knowing the vertex allows us to sketch the parabola accurately, as it defines the parabola's turning point. The symmetry of the parabola around the axis of symmetry (which passes through the vertex) further simplifies the graphing process. By identifying the vertex, we have determined the most crucial point for understanding the behavior of the quadratic function. The vertex, combined with the x-intercepts, forms the backbone of the parabola's graph, allowing us to visualize and analyze the function effectively. Understanding how to find the vertex is an essential skill for anyone studying quadratic functions and their applications.

Now that we have identified the _x-intercepts ((-1, 0) and (5, 0))* and the vertex (2, -9), we have enough information to sketch the graph of the quadratic function f(x) = (x+1)(x-5). The first step is to plot these points on the coordinate plane. Plot the x-intercepts (-1, 0) and (5, 0), which show where the parabola crosses the x-axis. Then, plot the vertex (2, -9), which represents the minimum point of the parabola since the coefficient of the x² term is positive (indicating the parabola opens upwards). With these three points plotted, we can begin to sketch the parabola. Recall that a parabola is a symmetrical U-shaped curve. The vertex is the turning point of the parabola, and the axis of symmetry (which is the vertical line x = 2 in this case) divides the parabola into two mirror images. Starting from the vertex (2, -9), draw a smooth curve that passes through the x-intercepts (-1, 0) and (5, 0). The curve should be symmetrical about the axis of symmetry. As you move away from the vertex, the parabola will continue to open upwards, increasing in both the positive and negative x-directions. To enhance the accuracy of the graph, you can plot additional points. For instance, you can choose x-values other than the x-intercepts and the vertex, substitute them into the function f(x) = (x+1)(x-5), and calculate the corresponding y-values. Plot these additional points and use them as guides for sketching the curve. The more points you plot, the more precise your graph will be. Sketching the graph of a quadratic function involves connecting the plotted points with a smooth curve, ensuring that the parabola opens in the correct direction and maintains its symmetrical shape. By plotting key points such as the x-intercepts and the vertex, and by utilizing the concept of symmetry, we can create an accurate representation of the quadratic function on the coordinate plane. This graphical representation provides valuable insights into the behavior and characteristics of the function.

To gain a comprehensive understanding of the quadratic function f(x) = (x+1)(x-5), it is essential to determine its domain and range. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For quadratic functions, the domain is typically all real numbers, as there are no restrictions on the values that x can take. In other words, you can substitute any real number into the function f(x) = (x+1)(x-5), and the result will be a real number. Therefore, the domain of this quadratic function is (-∞, ∞), which includes all real numbers. The range of a function, on the other hand, is the set of all possible output values (y-values) that the function can produce. The range is influenced by the vertex of the parabola and the direction in which the parabola opens. Since the parabola for the function f(x) = (x+1)(x-5) opens upwards (because the coefficient of the x² term is positive), the vertex represents the minimum point of the function. The y-coordinate of the vertex, which we found to be -9, is the minimum y-value that the function can take. The function will produce all y-values greater than or equal to -9. Therefore, the range of the function is [-9, ∞), which includes all real numbers greater than or equal to -9. Determining the domain and range provides a complete picture of the function's behavior. The domain tells us what input values are permissible, and the range tells us the possible output values. Together, they describe the extent of the function's reach on the coordinate plane. Understanding the domain and range is particularly crucial when applying quadratic functions to real-world scenarios, as it helps in interpreting the function's results within a meaningful context. For instance, in physics or engineering, the domain might represent the possible values of time, and the range might represent the possible values of distance or height. Analyzing the domain and range is an essential step in the process of graphing and understanding quadratic functions.

In this comprehensive guide, we have walked through the process of graphing the quadratic function f(x) = (x+1)(x-5), emphasizing the key steps and concepts involved. We began by identifying the *_x-intercepts_, which are the points where the parabola crosses the x-axis. These points provide a foundation for understanding the function's behavior and sketching its graph. Next, we found the midpoint between the x-intercepts, which helped us determine the axis of symmetry and the x-coordinate of the vertex. The vertex, representing the minimum or maximum point of the parabola, is a crucial feature for graphing quadratic functions. By evaluating the function at the x-coordinate of the vertex, we found the y-coordinate, thereby pinpointing the vertex's location. With the x-intercepts and the vertex identified, we proceeded to plot these points on the coordinate plane and sketch the parabola. The symmetrical U-shape of the parabola, along with the axis of symmetry, guided the sketching process, ensuring an accurate representation of the function. Additionally, we discussed how plotting additional points can further enhance the precision of the graph. Finally, we determined the domain and range of the function, which provide a complete picture of the function's behavior and the extent of its reach on the coordinate plane. The domain encompasses all possible input values, while the range encompasses all possible output values. Understanding the domain and range is essential for interpreting the function's results within a meaningful context. By mastering the steps and concepts outlined in this guide, you are now equipped with the necessary tools to graph quadratic functions effectively. The ability to graph quadratic functions is a valuable skill in mathematics and various fields, enabling you to analyze and interpret real-world phenomena that can be modeled by quadratic relationships. This comprehensive understanding will serve as a strong foundation for tackling more complex mathematical problems and applications.