Graphing The Quadratic Equation G(x) = -1/2(x+2)(x+8)

In this article, we will delve into the process of graphing the quadratic equation g(x) = -1/2(x+2)(x+8). Quadratic equations, characterized by their parabolic shape, play a significant role in various fields, including physics, engineering, and economics. Understanding how to graph these equations is crucial for visualizing their behavior and extracting valuable information about their properties. We will explore the key features of this particular equation, such as its intercepts, vertex, and axis of symmetry, and utilize these features to accurately sketch its graph. By the end of this discussion, you will have a comprehensive understanding of how to graph quadratic equations in factored form and be able to apply this knowledge to other similar problems.

Before we dive into graphing the specific equation, let's establish a solid understanding of quadratic equations in general. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic equation 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, and when a is negative, the parabola opens downwards. The vertex of the parabola is the point where the curve changes direction, and it represents either the minimum value (if the parabola opens upwards) or the maximum value (if the parabola opens downwards) of the quadratic function. The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. Understanding these basic properties is crucial for effectively graphing quadratic equations. Mastering these concepts will not only aid in visualizing the equation g(x) = -1/2(x+2)(x+8) but also in analyzing any quadratic function you encounter.

The equation we're focusing on, g(x) = -1/2(x+2)(x+8), is presented in factored form. This form provides valuable insights into the roots, or x-intercepts, of the equation. The roots are the values of x for which g(x) = 0, and they are the points where the parabola intersects the x-axis. In factored form, the roots are easily identified as the values that make each factor equal to zero. In this case, the factors are (x+2) and (x+8). Setting each factor to zero, we find that the roots are x = -2 and x = -8. These points are crucial for sketching the graph because they define where the parabola crosses the x-axis. The coefficient -1/2 plays a significant role in determining the shape and orientation of the parabola. Since it is negative, the parabola opens downwards, indicating that the vertex will be the maximum point. The magnitude of the coefficient also affects the width of the parabola; a smaller magnitude results in a wider parabola, while a larger magnitude results in a narrower parabola. Therefore, the negative coefficient (-1/2) in g(x) tells us that the parabola will open downwards and be wider than the standard parabola y = x². Understanding how the factored form and the leading coefficient affect the graph is essential for efficiently sketching the parabola. By recognizing the roots and the direction of opening, we can start to visualize the general shape and position of the graph.

To graph the equation g(x) = -1/2(x+2)(x+8) accurately, we need to determine its intercepts. Intercepts are the points where the graph intersects the coordinate axes. The x-intercepts, as we discussed earlier, are the points where the graph crosses the x-axis, and they occur when g(x) = 0. From the factored form of the equation, we have already identified the x-intercepts as x = -2 and x = -8. These points are crucial because they give us two fixed points on the parabola. The y-intercept is the point where the graph crosses the y-axis, and it occurs when x = 0. To find the y-intercept, we substitute x = 0 into the equation: g(0) = -1/2(0+2)(0+8) = -1/2(2)(8) = -8. Therefore, the y-intercept is (0, -8). The intercepts, both x and y, provide valuable reference points for sketching the graph. By plotting these points on the coordinate plane, we can start to visualize the overall shape and position of the parabola. The x-intercepts help define the horizontal spread of the parabola, while the y-intercept gives us a sense of its vertical position. Accurately identifying the intercepts is a crucial step in graphing any quadratic equation, as they serve as anchors for the curve. Locating these intercepts is essential for creating an accurate representation of the parabola on the coordinate plane.

The vertex is a critical point on a parabola, representing either its maximum or minimum value. For the equation g(x) = -1/2(x+2)(x+8), since the coefficient of the x² term is negative, the parabola opens downwards, and the vertex represents the maximum point. The x-coordinate of the vertex lies exactly midway between the two x-intercepts. In this case, the x-intercepts are x = -2 and x = -8. To find the x-coordinate of the vertex, we calculate the average of the x-intercepts: x_vertex = (-2 + -8) / 2 = -5. Now, to find the y-coordinate of the vertex, we substitute x = -5 into the equation: g(-5) = -1/2(-5+2)(-5+8) = -1/2(-3)(3) = 9/2 = 4.5. Therefore, the vertex of the parabola is (-5, 4.5). The vertex is crucial because it defines the turning point of the parabola and provides a reference for its vertical position. By knowing the vertex, we can accurately sketch the shape of the parabola and understand its maximum value. The axis of symmetry, which is a vertical line passing through the vertex, is also defined by the x-coordinate of the vertex. In this case, the axis of symmetry is the line x = -5. Precisely determining the vertex is paramount to accurately graphing the parabola, as it serves as the anchor point around which the rest of the curve is drawn.

Now that we have identified the key features of the equation g(x) = -1/2(x+2)(x+8), we can proceed to sketch its graph. We have found the x-intercepts to be (-2, 0) and (-8, 0), the y-intercept to be (0, -8), and the vertex to be (-5, 4.5). The axis of symmetry is the vertical line x = -5. To begin sketching, we plot these points on the coordinate plane. The x-intercepts give us two points where the parabola crosses the x-axis, the y-intercept gives us a point where the parabola crosses the y-axis, and the vertex gives us the maximum point of the parabola. Since the parabola opens downwards, we know that the curve will rise from the left x-intercept, reach its maximum at the vertex, and then fall towards the right x-intercept. Using the axis of symmetry as a guide, we can sketch a smooth curve that passes through the plotted points and is symmetrical about the line x = -5. The shape of the parabola should be consistent with the coefficient of the x² term, which is negative, indicating a downward-opening parabola. We should also note that the parabola is wider than the standard parabola y = x² due to the coefficient of -1/2. By carefully plotting the intercepts, vertex, and considering the axis of symmetry, we can accurately sketch the graph of the quadratic equation. Combining these elements allows for a clear and concise visualization of the function's behavior.

In this article, we have thoroughly explored the process of graphing the quadratic equation g(x) = -1/2(x+2)(x+8). We began by understanding the general properties of quadratic equations and their parabolic graphs. We then analyzed the specific equation, identifying its factored form and the significance of its coefficients. We determined the x-intercepts, y-intercept, and the vertex, which are crucial points for sketching the graph. Finally, we used these key features to accurately sketch the parabola, taking into account its direction of opening and width. The ability to graph quadratic equations is a valuable skill in mathematics and its applications. By understanding the relationships between the equation's form, its key features, and its graphical representation, we can effectively visualize and analyze quadratic functions. This knowledge empowers us to solve a wide range of problems in various fields, from physics and engineering to economics and finance. Mastering the techniques discussed here will undoubtedly enhance your understanding of quadratic equations and their applications. The comprehensive approach outlined in this article provides a solid foundation for further exploration of quadratic functions and their role in diverse mathematical and real-world contexts. By practicing these methods, you can confidently graph quadratic equations and interpret their graphical representations.