Graphing G(x) = -x² A Comprehensive Guide

In this comprehensive guide, we will delve into the process of graphing the quadratic function g(x) = -x². This function is a fundamental example in algebra and calculus, and understanding its graph is crucial for grasping more complex mathematical concepts. We will explore the key features of the graph, including its shape, vertex, axis of symmetry, and intercepts. This step-by-step explanation is designed to help students, educators, and anyone interested in mathematics gain a thorough understanding of how to visualize and interpret this function.

Quadratic functions are polynomial functions of the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards. The coefficient a plays a crucial role in determining the direction of the parabola. If a > 0, the parabola opens upwards, and if a < 0, the parabola opens downwards. The vertex of the parabola is the point where the curve changes direction, and it represents either the minimum (if a > 0) or maximum (if a < 0) value of the function. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.

In our specific case, g(x) = -x², we have a = -1, b = 0, and c = 0. Since a is negative, the parabola opens downwards. This means the vertex will be the highest point on the graph, representing the maximum value of the function. Understanding these basic properties of quadratic functions is essential for accurately graphing g(x) = -x² and similar functions. This detailed exploration will not only help in visualizing the function but also in understanding its behavior and applications in various mathematical contexts.

To accurately graph the function g(x) = -x², we will follow a systematic approach, breaking down the process into several key steps. This method ensures a clear understanding of the function's behavior and allows for precise plotting of its graph. By following these steps, you can confidently graph g(x) = -x² and similar quadratic functions.

1. Identify the Key Features

First, let's identify the key features of the function g(x) = -x². As we discussed earlier, this is a quadratic function with the form f(x) = ax² + bx + c, where a = -1, b = 0, and c = 0. Since a = -1, the parabola opens downwards, indicating that the vertex will be the maximum point. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The intercepts are the points where the graph intersects the x-axis and y-axis. Identifying these features will guide us in accurately sketching the graph. This initial step is crucial for understanding the overall shape and orientation of the parabola, allowing us to proceed with plotting specific points.

2. Determine the Vertex

The vertex of a parabola is the point where it changes direction, and it is a critical feature for graphing. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by the formula x = -b / 2a. In our case, g(x) = -x², we have a = -1 and b = 0. Plugging these values into the formula, we get x = -0 / (2 * -1) = 0. This means the x-coordinate of the vertex is 0. To find the y-coordinate, we substitute x = 0 into the function: g(0) = -(0)² = 0. Therefore, the vertex of the parabola is the point (0, 0). This point is also the origin, which simplifies our graphing process. Knowing the vertex helps us establish the central point around which the parabola is symmetrical, making it easier to plot the remaining points.

3. Find 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. The equation of the axis of symmetry is given by x = h, where h is the x-coordinate of the vertex. In our case, the vertex of g(x) = -x² is (0, 0), so the x-coordinate is 0. Therefore, the axis of symmetry is the vertical line x = 0, which is the y-axis. This axis acts as a mirror, reflecting the points on one side of the parabola onto the other side. Understanding the axis of symmetry is crucial for ensuring that the graph is symmetrical and accurate. It also simplifies the graphing process, as we only need to plot points on one side of the axis and then reflect them to the other side.

4. Calculate the Intercepts

Intercepts are the points where the graph of the function intersects the x-axis and y-axis. To find the x-intercepts, we set g(x) = 0 and solve for x: 0 = -x². This equation has only one solution, x = 0. Thus, the x-intercept is the point (0, 0), which is also the vertex. To find the y-intercept, we set x = 0 and evaluate g(0): g(0) = -(0)² = 0. Therefore, the y-intercept is also the point (0, 0). In this case, the x-intercept and y-intercept are the same, which is the vertex. This simplifies our graph, as we have a key point already identified. Knowing the intercepts provides important reference points for sketching the parabola and ensuring it accurately represents the function.

5. Plot Additional Points

To create a more accurate graph, we need to plot additional points on either side of the vertex. We can choose several x-values and calculate the corresponding y-values using the function g(x) = -x². For example, let's choose x = 1, 2, -1, and -2. For x = 1, g(1) = -(1)² = -1, so we have the point (1, -1). For x = 2, g(2) = -(2)² = -4, giving us the point (2, -4). For x = -1, g(-1) = -(-1)² = -1, so we have the point (-1, -1). For x = -2, g(-2) = -(-2)² = -4, giving us the point (-2, -4). Plotting these points, along with the vertex, provides a clearer picture of the parabola's shape. The more points we plot, the more accurate our graph will be. This step is essential for capturing the curve's behavior and ensuring a precise representation of the function.

6. Sketch the Graph

Now that we have identified the vertex (0, 0), the axis of symmetry x = 0, the intercepts (0, 0), and several additional points such as (1, -1), (2, -4), (-1, -1), and (-2, -4), we can sketch the graph of g(x) = -x². Starting from the vertex, draw a smooth curve that passes through the plotted points. Remember that the parabola opens downwards because the coefficient a is negative. The graph should be symmetrical about the y-axis (the axis of symmetry). Ensure that the curve is smooth and continuous, accurately representing the function's behavior. This final step brings together all the information we have gathered, allowing us to visualize the function in its entirety. A well-sketched graph provides a clear understanding of the function's properties and its relationship to the coordinate plane.

Once the graph of g(x) = -x² is sketched, we can analyze it to gain further insights into the function's properties and behavior. This analysis helps in understanding the function's characteristics and its applications in various mathematical contexts.

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 the quadratic function g(x) = -x², there are no restrictions on the input values, as any real number can be squared and then negated. Therefore, the domain of g(x) is all real numbers, which can be written as (-∞, ∞) in interval notation. The range of a function is the set of all possible output values (y-values) that the function can produce. Since the parabola opens downwards and the vertex is at (0, 0), the maximum value of g(x) is 0. All other y-values are negative. Therefore, the range of g(x) is all real numbers less than or equal to 0, which can be written as (-∞, 0] in interval notation. Understanding the domain and range provides a comprehensive view of the function's input and output values, helping to interpret its behavior within specific intervals.

Increasing and Decreasing Intervals

To determine the increasing and decreasing intervals of the function g(x) = -x², we examine the graph from left to right. A function is increasing in an interval if its y-values increase as x-values increase, and it is decreasing if its y-values decrease as x-values increase. For g(x) = -x², the graph increases from negative infinity up to the vertex at x = 0. Therefore, the function is increasing on the interval (-∞, 0). After the vertex, the graph decreases as x-values increase. Thus, the function is decreasing on the interval (0, ∞). Identifying the increasing and decreasing intervals helps in understanding how the function's output changes in relation to its input, providing valuable insights into its dynamic behavior.

Maximum and Minimum Values

The maximum and minimum values of a function are the highest and lowest points on its graph, respectively. For the quadratic function g(x) = -x², since the parabola opens downwards, it has a maximum value at its vertex. The vertex is (0, 0), so the maximum value of g(x) is 0. There is no minimum value for this function, as the parabola extends downwards indefinitely. The function approaches negative infinity as x moves away from 0 in either direction. Thus, g(x) has a maximum value of 0 at x = 0 and no minimum value. Understanding the maximum and minimum values helps in determining the extreme points of the function's output, which is crucial in various applications such as optimization problems.

In conclusion, graphing the function g(x) = -x² involves a systematic process of identifying key features, plotting points, and sketching the curve. We began by understanding the general form of quadratic functions and how the coefficient a affects the parabola's orientation. We then determined the vertex, axis of symmetry, and intercepts, which are crucial for accurately plotting the graph. By calculating additional points and connecting them with a smooth curve, we created a visual representation of the function. Finally, we analyzed the graph to understand its domain, range, increasing and decreasing intervals, and maximum and minimum values. This comprehensive approach provides a thorough understanding of the function g(x) = -x² and its graphical representation. By mastering this process, you can confidently graph and analyze other quadratic functions, gaining a deeper appreciation for their mathematical properties and applications.

This step-by-step guide not only helps in graphing g(x) = -x² but also lays the foundation for understanding more complex mathematical concepts. The ability to visualize functions and analyze their graphs is a valuable skill in mathematics and related fields. Whether you are a student learning algebra or a professional using mathematical models, a solid understanding of graphing techniques will undoubtedly enhance your problem-solving abilities and analytical skills.