Graphing Y=-x²+5x-6: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of quadratic equations and how to graph them. Specifically, we're going to tackle the equation y = -x² + 5x - 6. Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step, making it super easy to follow along. Graphing quadratic equations is a fundamental skill in algebra, and mastering it opens doors to understanding various real-world phenomena, from the trajectory of a ball to the design of suspension bridges. In this guide, we'll explore a straightforward method using a table of values to plot points and reveal the beautiful curve known as a parabola. So, grab your pencils, graph paper (or your favorite digital graphing tool), and let's get started on this exciting mathematical journey! Understanding the shape and behavior of quadratic equations is crucial for various applications in science, engineering, and even economics. By the end of this article, you'll not only be able to graph y = -x² + 5x - 6 but also have a solid foundation for graphing other quadratic equations.

1. Understanding the Basics of Quadratic Equations

Before we jump into graphing, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is y = ax² + bx + c, where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic equation is a parabola, a symmetrical U-shaped curve. This U-shape can open upwards or downwards, depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. Our equation, y = -x² + 5x - 6, fits this general form, with a = -1, b = 5, and c = -6. Since a is negative, we know our parabola will open downwards. Understanding these basic characteristics is the first step in effectively graphing quadratic functions. Recognizing the coefficients a, b, and c and their impact on the parabola's shape and direction is key to predicting the graph's overall appearance. This foundational knowledge will help us select appropriate values for our table and accurately plot the points. By understanding the fundamental properties of quadratic equations, we can approach graphing with confidence and gain a deeper appreciation for the mathematical relationships they represent.

2. Creating a Table of Values

The key to graphing y = -x² + 5x - 6 is to create a table of values. This table will help us find several points that lie on the parabola. To do this, we'll choose a range of x-values and then calculate the corresponding y-values using our equation. A good starting point is to choose x-values around the vertex of the parabola. The vertex is the highest or lowest point on the parabola, and it's a crucial point for graphing. We can find the x-coordinate of the vertex using the formula x = -b / 2a. In our case, a = -1 and b = 5, so the x-coordinate of the vertex is x = -5 / (2 * -1) = 2.5. Let's choose x-values around 2.5, such as 0, 1, 2, 2.5, 3, 4, and 5. Now, we'll plug each of these x-values into our equation y = -x² + 5x - 6 to find the corresponding y-values. For example, when x = 0, y = -(0)² + 5(0) - 6 = -6. When x = 1, y = -(1)² + 5(1) - 6 = -2. And so on. This systematic approach ensures that we capture the key features of the parabola, including its vertex and symmetry. By carefully selecting our x-values, we can efficiently generate a set of points that accurately represent the graph of the quadratic equation. Creating a table of values is a fundamental technique for graphing functions, and it provides a clear and organized way to connect the algebraic equation to its visual representation.

Table of Values for y = -x² + 5x - 6

3. Plotting the Points

Now that we have our table of values, it's time to plot these points on a graph. Each row in our table represents a coordinate point (x, y). For example, the first row (0, -6) gives us the point (0, -6), and the second row (1, -2) gives us the point (1, -2). On your graph paper (or digital graphing tool), draw your x and y axes. Then, carefully plot each of the points from our table. Make sure to scale your axes appropriately so that all your points fit on the graph. Plotting points accurately is crucial for obtaining the correct shape of the parabola. Double-check each point to ensure it's placed in the right location on the coordinate plane. Once you've plotted all the points from your table, you'll start to see the characteristic U-shape of the parabola emerge. The more points you plot, the clearer the curve will become. This visual representation of the equation helps us understand the relationship between the x and y values and how they change as x varies. Plotting the points is a fundamental step in graphing any function, and it provides a direct link between the numerical data in the table and the visual form of the graph.

4. Drawing the Parabola

With our points plotted, the final step is to draw the parabola. Remember, a parabola is a smooth, U-shaped curve. Connect the points you've plotted with a smooth curve, making sure the curve passes through all the points. The curve should be symmetrical about the vertical line that passes through the vertex (the point (2.5, 0.25) in our case). Since a is negative in our equation, the parabola opens downwards, meaning the vertex is the highest point on the curve. Extend the curve beyond the points you've plotted to show the general shape of the parabola. A well-drawn parabola accurately represents the quadratic equation and provides valuable insights into its behavior. The symmetry of the parabola is a key characteristic, and ensuring that your curve is symmetrical about the vertex is essential. Drawing a smooth, continuous curve requires a bit of practice, but with patience and attention to detail, you can create an accurate and visually appealing graph. The graph is a powerful tool for understanding the quadratic equation, allowing us to visualize its solutions, its vertex, and its overall behavior.

5. Key Features of the Graph

Let's take a closer look at our graph and identify some key features.

• Vertex: As we mentioned earlier, the vertex is the highest point on our parabola, which is (2.5, 0.25). The vertex represents the maximum value of the function.
• Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. For our parabola, the axis of symmetry is the line x = 2.5. This line divides the parabola into two symmetrical halves.
• X-intercepts: The x-intercepts are the points where the parabola intersects the x-axis (where y = 0). From our table and graph, we can see that the x-intercepts are (2, 0) and (3, 0). These are also the solutions to the equation * -x² + 5x - 6 = 0*.
• Y-intercept: The y-intercept is the point where the parabola intersects the y-axis (where x = 0). From our table and graph, we see that the y-intercept is (0, -6).

Identifying these key features provides a comprehensive understanding of the quadratic equation and its graph. The vertex gives us the maximum or minimum value of the function, the axis of symmetry highlights the parabola's symmetry, and the intercepts reveal the points where the parabola crosses the axes. These features are not only important for graphing but also for solving real-world problems involving quadratic relationships. Understanding the key features of the graph allows us to extract valuable information about the equation and its behavior.

6. Practice and Further Exploration

Congrats! You've successfully graphed the quadratic equation y = -x² + 5x - 6. The best way to master graphing quadratic equations is through practice. Try graphing other quadratic equations using the same method. You can vary the coefficients a, b, and c to see how they affect the shape and position of the parabola. For example, try graphing y = x² - 4x + 3 or y = 2x² + x - 1. You can also explore different methods for graphing quadratic equations, such as using the vertex form or factoring. Don't hesitate to use online graphing calculators or software to check your work and visualize the graphs. Graphing quadratic equations is a fundamental skill in algebra, and the more you practice, the more confident you'll become. Experiment with different equations, explore different graphing techniques, and challenge yourself to understand the relationships between the equation and its graph. The journey of mathematical discovery is continuous, and there's always more to learn and explore.

By mastering the techniques discussed in this guide, you've taken a significant step towards understanding quadratic functions and their graphs. Keep practicing, keep exploring, and you'll be graphing parabolas like a pro in no time!