Completing The Table And Graphing The Quadratic Relation Y = 2x² - 5x + 1
This article will guide you through the process of completing a table of values for the quadratic relation y = 2x² - 5x + 1, and then using that table to draw the graph of the relation. We'll also explore how to choose an appropriate scale for the graph and use it to accurately represent the relationship between x and y. Understanding quadratic relations is crucial in mathematics as they appear in various real-world scenarios, such as projectile motion and optimization problems. Let's dive in and master this concept.
(a) Completing the Table of Values
To complete the table, we need to substitute each given value of x into the equation y = 2x² - 5x + 1 and calculate the corresponding value of y. This process involves careful arithmetic and attention to the order of operations (PEMDAS/BODMAS). For each x value, we'll perform the squaring operation first, then multiplication, and finally addition and subtraction. This step-by-step approach ensures accuracy and minimizes errors. Let's begin by calculating the y values for the missing x values in the table.
Detailed Calculations for Each x-value
- When x = -3:
- Substitute x = -3 into the equation: y = 2(-3)² - 5(-3) + 1
- Calculate the square: y = 2(9) - 5(-3) + 1
- Multiply: y = 18 + 15 + 1
- Add: y = 34
- Therefore, when x = -3, y = 34.
- When x = 1:
- Substitute x = 1 into the equation: y = 2(1)² - 5(1) + 1
- Calculate the square: y = 2(1) - 5(1) + 1
- Multiply: y = 2 - 5 + 1
- Add and subtract: y = -2
- Therefore, when x = 1, y = -2.
- When x = 2:
- Substitute x = 2 into the equation: y = 2(2)² - 5(2) + 1
- Calculate the square: y = 2(4) - 5(2) + 1
- Multiply: y = 8 - 10 + 1
- Add and subtract: y = -1
- Therefore, when x = 2, y = -1.
- When x = 3:
- Substitute x = 3 into the equation: y = 2(3)² - 5(3) + 1
- Calculate the square: y = 2(9) - 5(3) + 1
- Multiply: y = 18 - 15 + 1
- Add and subtract: y = 4
- Therefore, when x = 3, y = 4.
- When x = 4:
- Substitute x = 4 into the equation: y = 2(4)² - 5(4) + 1
- Calculate the square: y = 2(16) - 5(4) + 1
- Multiply: y = 32 - 20 + 1
- Add and subtract: y = 13
- Therefore, when x = 4, y = 13.
Completed Table
Now that we have calculated all the missing y values, we can complete the table:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|
| y | 34 | 8 | 1 | -1 | -2 | -1 | 4 | 13 | 26 |
This completed table provides us with a set of coordinate pairs (x, y) that we can use to plot the graph of the quadratic relation. Each pair represents a point on the curve, and by connecting these points, we can visualize the shape and behavior of the quadratic function. Understanding how to complete such tables is essential for graphing and analyzing various mathematical functions, not just quadratic ones.
(b) Graphing the Quadratic Relation
Now that we have the completed table of values, the next step is to graph the quadratic relation y = 2x² - 5x + 1. This involves choosing an appropriate scale for the axes, plotting the points from the table, and then drawing a smooth curve through the points. The scale is crucial as it determines how the graph is represented on the coordinate plane. A well-chosen scale will make the graph clear, easy to read, and accurately reflect the relationship between x and y.
Choosing an Appropriate Scale
The scale given is 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis. This means that every 2 centimeters on the x-axis represents a change of 1 in the value of x, and every 2 centimeters on the y-axis represents a change of 5 in the value of y. This scale is chosen to accommodate the range of x and y values in our table, ensuring that the graph fits comfortably on the paper and that the points are reasonably spaced apart.
- X-axis: The x values range from -3 to 5. Using a scale of 2cm to 1 unit allows us to represent these values clearly. For instance, x = -3 would be plotted 6cm to the left of the origin (0), and x = 5 would be plotted 10cm to the right of the origin.
- Y-axis: The y values range from -2 to 34. With a scale of 2cm to 5 units, we can effectively represent this range. The value y = -2 would be plotted 0.8cm below the origin, and y = 34 would be plotted 13.6cm above the origin. This scale allows us to see the overall shape of the parabola without compressing the graph too much.
Plotting the Points and Drawing the Curve
Using the chosen scale, we can now plot the points from our completed table onto the graph paper. Each point corresponds to an (x, y) pair. For example, the point (-3, 34) would be located 6cm to the left of the origin and 13.6cm above the origin. Similarly, the point (0, -1) would be located at the origin horizontally and 0.4cm below the origin vertically. Once all the points are plotted, we can draw a smooth curve through them. This curve should be a parabola, which is the characteristic shape of a quadratic function. The curve should pass as close as possible to all the plotted points, and it should be symmetrical about its axis of symmetry.
Interpreting the Graph
The resulting graph visually represents the relationship between x and y as defined by the quadratic equation. We can observe the following features of the graph:
- Parabola: The U-shaped curve indicates that we are dealing with a quadratic function.
- Vertex: The vertex is the lowest point on the parabola (in this case, since the coefficient of x² is positive). It represents the minimum value of the function. By looking at the graph, we can estimate the coordinates of the vertex.
- Axis of Symmetry: This is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. We can identify the equation of the axis of symmetry from the graph.
- Y-intercept: This is the point where the parabola intersects the y-axis. It corresponds to the value of y when x = 0. From the graph (and the table), we know the y-intercept is -1.
- X-intercepts (Roots): These are the points where the parabola intersects the x-axis. They correspond to the values of x when y = 0. We can estimate the x-intercepts from the graph. The accuracy of our graph directly impacts the precision with which we can determine these key features.
By carefully plotting the points and drawing the curve, we can obtain a clear and accurate representation of the quadratic relation. This graphical representation allows us to visualize the behavior of the function and extract important information about its properties. Understanding how to graph quadratic relations is fundamental for solving quadratic equations and understanding their applications in various fields.
Conclusion
In this article, we have successfully completed the table of values for the quadratic relation y = 2x² - 5x + 1 and graphed the relation using an appropriate scale. We emphasized the importance of accurate calculations and careful plotting to obtain a clear and informative graph. The ability to complete tables of values and graph quadratic relations is a valuable skill in mathematics, providing a visual understanding of the relationship between variables and enabling us to solve a wide range of problems. By mastering these concepts, you'll be well-equipped to tackle more advanced topics in algebra and calculus. Remember, practice is key, so continue working on similar problems to solidify your understanding and build your confidence.
