Graphing The Quadratic Function Y=(1/2)x^2+2x-8 A Step-by-Step Guide
In mathematics, graphing functions is a fundamental skill. It allows us to visualize the behavior of a function and understand its properties. When you graph a function, you create a visual representation of the relationship between the input values (x) and the output values (y). By plotting these points on a coordinate plane, you can see the shape of the function and identify key features such as intercepts, turning points, and asymptotes. Graphing is not just about plotting points; it's about understanding the underlying function and its characteristics. It enables us to solve equations, analyze data, and make predictions based on mathematical models. In this comprehensive guide, we will delve into the process of graphing the quadratic function y = (1/2)x² + 2x - 8. This function is a parabola, a U-shaped curve that is a common sight in algebra and calculus. Mastering the techniques to graph quadratic functions will give you a solid foundation for more advanced mathematical concepts. We will explore various methods, including finding the vertex, intercepts, and using additional points to sketch an accurate graph. By the end of this guide, you will be equipped with the knowledge and skills to confidently graph quadratic functions and understand their significance in mathematical analysis. Understanding how to graph this equation will help us understand the behavior of quadratic functions, a critical concept in algebra and calculus. Let's explore step-by-step how to accurately graph this function.
Before diving into the graphing process, it’s crucial to understand the standard form of a quadratic function: y = ax² + bx + c. In our given function, y = (1/2)x² + 2x - 8, we can identify the coefficients as follows: a = 1/2, b = 2, and c = -8. The coefficient a plays a significant role in determining the shape and direction of the parabola. Specifically, the sign of a tells us whether the parabola opens upwards or downwards. If a is positive, the parabola opens upwards, creating a U-shape, and if a is negative, it opens downwards, creating an inverted U-shape. In our case, a = 1/2, which is positive, so the parabola will open upwards. The magnitude of a also affects the shape of the parabola. A smaller absolute value of a results in a wider parabola, while a larger absolute value makes the parabola narrower. In our function, a = 1/2 indicates that the parabola will be wider compared to a parabola with a = 1.
The vertex of a parabola is its turning point—the minimum point if the parabola opens upwards, or the maximum point if it opens downwards. The vertex is a crucial feature of a parabola, as it provides a reference point for graphing the function. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Once we have the x-coordinate, we can find the y-coordinate by substituting this value back into the original equation. This gives us the vertex in the form (x, y). The c coefficient, which is the constant term in the quadratic function, represents the y-intercept of the parabola. The y-intercept is the point where the parabola intersects the y-axis. It is found by setting x = 0 in the equation and solving for y. In our function, c = -8, so the y-intercept is at the point (0, -8). Understanding these basic components—the coefficients a, b, and c, the vertex, and the y-intercept—is essential for graphing the quadratic function accurately. By analyzing these elements, we can get a clear picture of the parabola's shape, position, and orientation on the coordinate plane, making the graphing process more straightforward and intuitive.
The vertex is a critical point for graphing any parabola, as it represents the minimum or maximum point of the curve. To find the vertex of the function y = (1/2)x² + 2x - 8, we will use the formula x = -b / 2a to determine the x-coordinate. Recall that in our equation, a = 1/2 and b = 2. Plugging these values into the formula, we get:
x = -2 / (2 * (1/2))
x = -2 / 1
x = -2
Thus, the x-coordinate of the vertex is -2. Now, to find the y-coordinate, we substitute this x-value back into the original equation:
y = (1/2)(-2)² + 2(-2) - 8
y = (1/2)(4) - 4 - 8
y = 2 - 4 - 8
y = -10
Therefore, the y-coordinate of the vertex is -10. Combining the x and y coordinates, we find that the vertex of the parabola is at the point (-2, -10). This point represents the lowest point on the parabola, as the coefficient a is positive, indicating that the parabola opens upwards. The vertex is not only a key point for graphing but also a crucial feature for understanding the behavior of the quadratic function. It provides the axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. This axis of symmetry helps in plotting additional points on the graph, as for every point on one side of the axis, there is a corresponding point on the other side at the same height. Knowing the vertex's location allows us to sketch the parabola more accurately, as it serves as a reference point around which the curve is shaped. It also helps in identifying the minimum or maximum value of the function, which can be essential in various applications, such as optimization problems in physics, engineering, and economics. In summary, finding the vertex is a crucial step in graphing quadratic functions, as it provides valuable information about the parabola's position, shape, and behavior. With the vertex at (-2, -10), we can proceed to find other key points and accurately sketch the graph.
To fully graph the function, we need to find the intercepts: the points where the parabola crosses the x and y axes. These intercepts provide additional reference points that help in sketching an accurate curve. The y-intercept is the point where the parabola intersects the y-axis. To find it, we set x = 0 in the equation y = (1/2)x² + 2x - 8:
y = (1/2)(0)² + 2(0) - 8
y = 0 + 0 - 8
y = -8
So, the y-intercept is the point (0, -8). This means the parabola crosses the y-axis at -8. The x-intercepts are the points where the parabola intersects the x-axis. To find them, we set y = 0 in the equation and solve for x: 0 = (1/2)x² + 2x - 8. This is a quadratic equation that we can solve using the quadratic formula, factoring, or completing the square. In this case, the quadratic formula is a suitable method. The quadratic formula is given by:
x = [-b ± √(b² - 4ac)] / 2a
In our equation, a = 1/2, b = 2, and c = -8. Plugging these values into the formula, we get:
x = [-2 ± √(2² - 4(1/2)(-8))] / (2(1/2))
x = [-2 ± √(4 + 16)] / 1
x = [-2 ± √20] / 1
x = -2 ± 2√5
Thus, the x-intercepts are x = -2 + 2√5 and x = -2 - 2√5. Approximating these values, we get x ≈ 2.47 and x ≈ -6.47. Therefore, the x-intercepts are approximately (2.47, 0) and (-6.47, 0). The x-intercepts are also known as the roots or zeros of the quadratic function. They are the points where the function's value is zero. The number of x-intercepts (two, one, or none) depends on the discriminant (the part under the square root in the quadratic formula, b² - 4ac). If the discriminant is positive, there are two distinct x-intercepts; if it's zero, there is one x-intercept (the vertex touches the x-axis); and if it's negative, there are no real x-intercepts. In our case, the discriminant is 20, which is positive, so we have two x-intercepts. Having found the y-intercept (0, -8) and the x-intercepts (2.47, 0) and (-6.47, 0), we now have three crucial points that will help us in accurately sketching the parabola. These intercepts, along with the vertex we found earlier, provide a clear framework for graphing the quadratic function.
While the vertex and intercepts provide a good foundation for graphing the parabola, plotting additional points can enhance the accuracy of the graph. This is especially useful for understanding the curve's shape more precisely. To plot additional points, we choose arbitrary x-values and substitute them into the function y = (1/2)x² + 2x - 8 to find the corresponding y-values. The points should be selected strategically, typically around the vertex and intercepts, to capture the curve's behavior. Considering the vertex is at (-2, -10), we can choose x-values such as -4, -1, -3, and 0. We already know the y-value for x = 0 (the y-intercept), so let’s calculate the y-values for the other points:
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For x = -4:
y = (1/2)(-4)² + 2(-4) - 8
y = (1/2)(16) - 8 - 8
y = 8 - 8 - 8
y = -8
So, the point is (-4, -8).
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For x = -1:
y = (1/2)(-1)² + 2(-1) - 8
y = (1/2)(1) - 2 - 8
y = 0.5 - 2 - 8
y = -9.5
So, the point is (-1, -9.5).
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For x = -3:
y = (1/2)(-3)² + 2(-3) - 8
y = (1/2)(9) - 6 - 8
y = 4.5 - 6 - 8
y = -9.5
So, the point is (-3, -9.5).
Now we have a few additional points: (-4, -8), (-1, -9.5), and (-3, -9.5). These points, combined with the vertex and intercepts, will give us a more detailed picture of the parabola's shape. The symmetry of the parabola can also be used to our advantage. Since parabolas are symmetrical about their vertex, for every point (x, y) on the parabola, there is a corresponding point at the same y-value on the other side of the vertex. The axis of symmetry is the vertical line passing through the vertex. In our case, the axis of symmetry is x = -2. This means that the point (-4, -8) is symmetrical to the point (0, -8) (the y-intercept), and the points (-1, -9.5) and (-3, -9.5) are symmetrical to each other. By plotting these additional points, we can sketch a more accurate and smooth curve for the parabola. This step is crucial in visualizing the function’s behavior and understanding its characteristics. The more points we plot, the more confident we can be in the accuracy of our graph. These additional points help to fill in the gaps between the key features and provide a comprehensive view of the parabola's shape and position on the coordinate plane.
With the vertex, intercepts, and additional points calculated, we are now ready to sketch the graph of the quadratic function y = (1/2)x² + 2x - 8. The first step is to draw the coordinate plane, including both the x and y axes. It’s important to choose an appropriate scale for the axes so that all the points we’ve calculated can be plotted clearly. Mark the x-axis and y-axis with evenly spaced intervals, ensuring that the range covers all the key points, including the vertex (-2, -10), the intercepts (-6.47, 0), (2.47, 0), and (0, -8), and the additional points (-4, -8), (-1, -9.5), and (-3, -9.5).
Next, plot all the points we’ve identified on the coordinate plane. Start with the vertex (-2, -10), which is the lowest point on the parabola since the parabola opens upwards. Then, plot the x-intercepts (-6.47, 0) and (2.47, 0), which are the points where the parabola crosses the x-axis. Plot the y-intercept (0, -8), where the parabola crosses the y-axis. Finally, plot the additional points (-4, -8), (-1, -9.5), and (-3, -9.5). These points help to define the curve’s shape more accurately.
Once all the points are plotted, carefully draw a smooth, U-shaped curve that passes through all the points. Remember that the parabola is symmetrical about the vertical line that passes through the vertex (the axis of symmetry). In our case, the axis of symmetry is the line x = -2. This means that the left and right sides of the parabola should be mirror images of each other. As you sketch the curve, ensure that it opens upwards since the coefficient a in the quadratic function is positive (a = 1/2). The curve should smoothly connect the points, with no sharp corners or abrupt changes in direction. Extend the curve beyond the plotted points to show the general shape of the parabola. A well-sketched graph should clearly show the vertex, intercepts, and the overall shape of the parabola. The accuracy of the graph depends on the precise plotting of points and the smooth connection between them. Visualizing the parabola on the coordinate plane provides a comprehensive understanding of the function’s behavior. From the graph, we can observe the minimum value of the function (the y-coordinate of the vertex), the range of the function (all y-values greater than or equal to -10), and the symmetry of the curve. Graphing the quadratic function is not just a mechanical process; it’s a visual representation that brings the equation to life and helps in understanding its properties and applications.
In this comprehensive guide, we have walked through the process of graphing the quadratic function y = (1/2)x² + 2x - 8 step by step. We began by understanding the basic form of a quadratic function and identifying the coefficients a, b, and c. We recognized that the coefficient a determines the direction in which the parabola opens, and in our case, since a is positive, the parabola opens upwards. Next, we found the vertex of the parabola using the formula x = -b / 2a to find the x-coordinate, and then substituted this value back into the equation to find the y-coordinate. This gave us the vertex at (-2, -10), a crucial point representing the minimum of the function. We then determined the intercepts, both the y-intercept and the x-intercepts. The y-intercept was found by setting x = 0, yielding the point (0, -8). The x-intercepts were found by setting y = 0 and solving the quadratic equation, resulting in two points, approximately (2.47, 0) and (-6.47, 0). To enhance the accuracy of the graph, we plotted additional points by choosing arbitrary x-values and calculating the corresponding y-values. This provided a clearer picture of the parabola’s shape and helped in creating a smoother curve. Finally, we sketched the graph by plotting all the key points—the vertex, intercepts, and additional points—on the coordinate plane and drawing a smooth, U-shaped curve through them. We emphasized the importance of symmetry about the vertex and the smooth connection between points to accurately represent the parabola.
Graphing a quadratic function is a fundamental skill in mathematics, providing a visual representation of the function’s behavior and properties. By following these steps, you can confidently graph any quadratic function and gain a deeper understanding of its characteristics. Understanding the graph of a quadratic function allows you to identify key features such as the minimum or maximum value (the vertex), the zeros of the function (x-intercepts), and the symmetry of the curve. These insights are invaluable in various applications, from solving optimization problems to analyzing real-world scenarios modeled by quadratic functions. This process is not just about plotting points; it’s about understanding the relationship between the equation and its graphical representation. By mastering the techniques outlined in this guide, you can apply these skills to more complex mathematical problems and gain a solid foundation in algebra and calculus. Whether you’re a student learning the basics of graphing or a professional using mathematical models, the ability to accurately graph quadratic functions is an essential tool in your mathematical toolkit.
