Graphing Y = -1/2x^2 - 1 Identify Vertex And Maximum

In this article, we will explore the process of graphing the quadratic equation y = -1/2x^2 - 1. We will delve into identifying the vertex, determining whether it represents a minimum or maximum point, and understanding the overall shape of the parabola. This comprehensive guide will provide you with a step-by-step approach to graphing quadratic equations, ensuring a clear understanding of the key concepts involved.

Understanding Quadratic Equations

Before we dive into graphing the specific equation, let's first establish a strong foundation by understanding the general form of quadratic equations and their properties. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is:

y = ax^2 + bx + c

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 either upwards or downwards. The coefficient a plays a crucial role in determining the shape and 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. 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, dividing the parabola into two symmetrical halves.

Analyzing the Equation y = -1/2x^2 - 1

Now, let's focus on the given equation: y = -1/2x^2 - 1. By comparing this equation to the standard form, we can identify the coefficients:

• a = -1/2
• b = 0
• c = -1

Since a = -1/2, which is less than 0, we know that the parabola opens downwards. This means the vertex will represent the maximum point of the graph. To find the vertex, we can use the vertex formula:

x-coordinate of vertex (h) = -b / 2a

In our case, b = 0 and a = -1/2, so:

h = -0 / (2 * -1/2) = 0

To find the y-coordinate of the vertex (k), we substitute the x-coordinate (h = 0) back into the equation:

k = -1/2(0)^2 - 1 = -1

Therefore, the vertex of the parabola is (0, -1). Since the parabola opens downwards, this vertex represents the maximum point of the graph.

Graphing the Parabola

To graph the parabola y = -1/2x^2 - 1, we can follow these steps:

1. Plot the vertex: We already found the vertex to be (0, -1). Plot this point on the coordinate plane. This point is the cornerstone of our graph, serving as the turning point of the parabola and the anchor for its symmetric shape. Understanding the vertex's significance is paramount in accurately depicting the parabola's trajectory and behavior across the coordinate plane.
2. Find additional points: To get a better understanding of the parabola's shape, we need to find a few more points. We can do this by choosing some x-values and plugging them into the equation to find the corresponding y-values. Remember that parabolas are symmetrical, so we can choose x-values on both sides of the vertex. Select x-values that are both positive and negative, and ideally, some that are equidistant from the vertex's x-coordinate. This symmetrical approach ensures that the plotted points accurately reflect the parabolic shape and contribute to a well-rounded graph. For example, we can choose x = -2, -1, 1, and 2.
   • For x = -2: y = -1/2(-2)^2 - 1 = -1/2(4) - 1 = -2 - 1 = -3. So, the point is (-2, -3).
   • For x = -1: y = -1/2(-1)^2 - 1 = -1/2(1) - 1 = -1/2 - 1 = -3/2. So, the point is (-1, -3/2).
   • For x = 1: y = -1/2(1)^2 - 1 = -1/2(1) - 1 = -1/2 - 1 = -3/2. So, the point is (1, -3/2).
   • For x = 2: y = -1/2(2)^2 - 1 = -1/2(4) - 1 = -2 - 1 = -3. So, the point is (2, -3).
3. Plot the additional points: Plot the points you found in the previous step on the coordinate plane. These points serve as crucial guides in shaping the parabola, ensuring that the curve accurately reflects the equation's behavior. By plotting these points, we gain a more comprehensive understanding of the parabola's width, steepness, and overall position on the coordinate plane. Each point acts as an anchor, contributing to a visually precise and informative graphical representation.
4. Draw the parabola: Now, connect the points with a smooth, U-shaped curve. Remember that the parabola is symmetrical, so the curve should be mirrored on both sides of the axis of symmetry (which is the vertical line passing through the vertex). As you sketch the parabola, pay close attention to maintaining its smooth, symmetrical shape, ensuring that it accurately reflects the quadratic equation's characteristics. This final step brings the graph to life, visually representing the relationship between the x and y values.

Key Features of the Graph

Here's a summary of the key features of the graph of y = -1/2x^2 - 1:

• Vertex: (0, -1)
• Maximum: The vertex represents the maximum point of the graph.
• Axis of Symmetry: x = 0 (the y-axis)
• Opens Downwards: The parabola opens downwards because the coefficient a is negative.

Conclusion

Graphing quadratic equations like y = -1/2x^2 - 1 involves understanding the standard form, identifying the coefficients, finding the vertex, and plotting additional points. By following these steps, you can accurately graph any quadratic equation and gain valuable insights into its properties. Remember, the vertex is a crucial point that determines the maximum or minimum value of the function, and the coefficient a dictates the direction and shape of the parabola. With practice, graphing quadratic equations will become a straightforward and insightful process.

Keywords: quadratic equations, graphing, vertex, maximum, parabola, axis of symmetry, coefficients, standard form, y = -1/2x^2 - 1