Graphing Y=-x^2-6x-8 Finding Roots And Vertex

In this article, we will explore how to graph the quadratic equation y = -x^2 - 6x - 8 and determine its roots. We will plot five key points, including the roots and the vertex, to accurately represent the parabola. By analyzing the graph, we can visually identify the roots of the equation -x^2 - 6x - 8 = 0. Understanding how to graph quadratic equations and find their roots is a fundamental concept in algebra, with applications in various fields such as physics, engineering, and economics.

Understanding Quadratic Equations

Before we dive into graphing, let's briefly discuss quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0. The roots of a quadratic equation are the x-values where the parabola intersects the x-axis, also known as the x-intercepts. These roots represent the solutions to the equation when y = 0. The vertex of the parabola is the point where the parabola changes direction, either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). The vertex plays a crucial role in understanding the behavior and symmetry of the quadratic function. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. By identifying the vertex, roots, and axis of symmetry, we can effectively graph and analyze quadratic equations.

Finding the Roots of the Equation

To begin, we need to find the roots of the equation -x^2 - 6x - 8 = 0. The roots of a quadratic equation are the values of x for which the equation equals zero. There are several methods to find the roots, including factoring, using the quadratic formula, or completing the square. In this case, we will use factoring, which is often the simplest method when applicable. Factoring involves expressing the quadratic equation as a product of two binomials. For the equation -x^2 - 6x - 8 = 0, we can factor out a -1 to simplify the process, resulting in -(x^2 + 6x + 8) = 0. Next, we look for two numbers that multiply to 8 and add up to 6. These numbers are 4 and 2. Thus, we can factor the quadratic expression as -(x + 4)(x + 2) = 0. Setting each factor equal to zero gives us the roots: x + 4 = 0 and x + 2 = 0. Solving these equations, we find that x = -4 and x = -2. Therefore, the roots of the equation are -4 and -2. These roots represent the points where the parabola intersects the x-axis, which are crucial for graphing the equation. The ability to find roots is fundamental in solving quadratic equations and understanding the behavior of the corresponding parabolas.

Determining the Vertex

Next, we need to determine the vertex of the parabola. The vertex is the point where the parabola reaches its maximum or minimum value. For a quadratic equation in the form y = ax^2 + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. In our equation, y = -x^2 - 6x - 8, a = -1 and b = -6. Plugging these values into the formula, we get x = -(-6) / (2 * -1) = 6 / -2 = -3. This x-coordinate tells us the vertical line where the vertex lies, which is the axis of symmetry. To find the y-coordinate of the vertex, we substitute this x-value back into the original equation: y = -(-3)^2 - 6(-3) - 8 = -9 + 18 - 8 = 1. Therefore, the vertex of the parabola is the point (-3, 1). Since the coefficient of the x^2 term is negative (a = -1), the parabola opens downwards, indicating that the vertex is the maximum point of the curve. Knowing the vertex is essential for accurately graphing the parabola, as it serves as the turning point and the axis of symmetry helps define the shape of the curve.

Plotting the Points and Graphing the Equation

Now that we have the roots and the vertex, we can plot these points on the coordinate plane. The roots are (-4, 0) and (-2, 0), and the vertex is (-3, 1). These three points provide a good starting point for graphing the parabola. However, to ensure accuracy, we need to plot two more points. We can choose x-values that are symmetrical around the vertex. For example, we can choose x = -1 and x = -5. When x = -1, y = -(-1)^2 - 6(-1) - 8 = -1 + 6 - 8 = -3, giving us the point (-1, -3). When x = -5, y = -(-5)^2 - 6(-5) - 8 = -25 + 30 - 8 = -3, giving us the point (-5, -3). These points are symmetrical around the vertex, as expected. Now, we have five points: (-4, 0), (-2, 0), (-3, 1), (-1, -3), and (-5, -3). Plotting these points on a coordinate plane and connecting them with a smooth curve will give us the graph of the parabola y = -x^2 - 6x - 8. The parabola opens downwards, with the vertex as the highest point, and is symmetrical around the vertical line x = -3. Graphing these points accurately is crucial for visualizing the quadratic equation and understanding its behavior.

Analyzing the Graph and Determining the Roots

By analyzing the graph of the equation y = -x^2 - 6x - 8, we can visually confirm the roots of the equation -x^2 - 6x - 8 = 0. The roots are the points where the parabola intersects the x-axis. From the graph, we can see that the parabola intersects the x-axis at x = -4 and x = -2. These are the x-values where y = 0, which are the solutions to the equation. The graph also provides a visual representation of the parabola's symmetry and the location of the vertex. The vertex, being the maximum point of the parabola, is at (-3, 1), and the parabola is symmetrical around the vertical line x = -3. The shape of the parabola and its position on the coordinate plane give us valuable insights into the behavior of the quadratic equation. Visualizing the roots and the vertex on the graph enhances our understanding of the solutions to the equation and the overall nature of the quadratic function. Therefore, graphical analysis is a powerful tool in understanding and solving quadratic equations.

Conclusion

In conclusion, we have successfully graphed the equation y = -x^2 - 6x - 8 by plotting five key points: the roots (-4, 0) and (-2, 0), and the vertex (-3, 1), along with two additional symmetrical points. By analyzing the graph, we visually confirmed that the roots of the equation -x^2 - 6x - 8 = 0 are -4 and -2. This process demonstrates the importance of understanding quadratic equations, their graphs, and the relationship between the roots and the vertex. Graphing quadratic equations is a fundamental skill in algebra, providing a visual representation of the equation's solutions and behavior. By mastering the techniques of finding roots, determining the vertex, and plotting points, we can effectively analyze and solve quadratic equations in various mathematical and real-world contexts.