Analyzing Integer Coordinates On Parabola Y=x²-3x+2 And Line Y=0
In the realm of mathematics, particularly in coordinate geometry, exploring the interplay between curves and lines often leads to fascinating insights. This article delves into a specific problem involving a parabola and a line, focusing on the points where both their x and y coordinates are integers. We are given two sets, A and B. Set A comprises points with integer coordinates that lie on the parabola defined by the equation y = x² - 3x + 2. Set B, on the other hand, consists of points with integer coordinates that lie on the line y = 0. Our objective is to comprehensively analyze these sets, identify their elements, and discuss the mathematical concepts involved.
To begin, let's formally define the sets A and B:
- Set A: A = (x, y)
- Set B: B = (x, y)
Set A encapsulates all points with integer coordinates that satisfy the parabolic equation y = x² - 3x + 2. This means we are looking for integer values of x that, when plugged into the equation, yield integer values for y. The parabola's equation is a quadratic equation, which means it forms a U-shaped curve when plotted on a coordinate plane. The integer points on this curve are where the curve intersects the grid lines of the coordinate plane at integer values.
Set B is simpler to understand. It represents all points with integer coordinates that lie on the x-axis, since the equation y = 0 defines the x-axis. Any point on the x-axis has a y-coordinate of 0, and we are only interested in points where the x-coordinate is also an integer.
To determine the elements of Set A, we need to find integer pairs (x, y) that satisfy the equation y = x² - 3x + 2. This involves substituting integer values for x and checking if the resulting y value is also an integer. The process requires a systematic approach, often involving factoring or completing the square to better understand the quadratic equation's behavior.
First, let's factor the quadratic equation: y = x² - 3x + 2 = (x - 1)(x - 2). This factored form provides valuable insights. We can see that y = 0 when x = 1 or x = 2. This gives us two points in Set A: (1, 0) and (2, 0). These points are significant because they represent the x-intercepts of the parabola.
Now, let's explore other integer values of x. If x = 0, then y = (0 - 1)(0 - 2) = 2, giving us the point (0, 2). If x = 3, then y = (3 - 1)(3 - 2) = 2, giving us the point (3, 2). Notice that the y-values are the same for x = 0 and x = 3 due to the symmetry of the parabola.
Let's consider negative integer values. If x = -1, then y = (-1 - 1)(-1 - 2) = (-2)(-3) = 6, giving us the point (-1, 6). If x = 4, then y = (4 - 1)(4 - 2) = (3)(2) = 6, giving us the point (4, 6). Again, we observe symmetry in the y-values.
We can continue this process for other integer values of x. As |x| increases, the value of y will generally increase as well, since the quadratic term x² dominates the equation. This means that the points will move further away from the x-axis as we move away from the vertex of the parabola. By calculating several points, we can see the pattern:
- x = -2, y = (-2 - 1)(-2 - 2) = (-3)(-4) = 12, point (-2, 12)
- x = 5, y = (5 - 1)(5 - 2) = (4)(3) = 12, point (5, 12)
- x = -3, y = (-3 - 1)(-3 - 2) = (-4)(-5) = 20, point (-3, 20)
- x = 6, y = (6 - 1)(6 - 2) = (5)(4) = 20, point (6, 20)
So far, we have identified the following points in Set A: (1, 0), (2, 0), (0, 2), (3, 2), (-1, 6), (4, 6), (-2, 12), (5, 12), (-3, 20), (6, 20). These points illustrate the nature of Set A as a collection of discrete points on the parabola where both coordinates are integers.
Finding the elements of Set B is much simpler. Set B consists of points on the line y = 0 where the x-coordinate is an integer. This is essentially the set of all integers on the x-axis. We can represent Set B as: B = (x, 0) . Some examples of points in Set B are: (-3, 0), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (3, 0), and so on.
An interesting question arises: What is the intersection of Sets A and B? The intersection, denoted as A ∩ B, consists of points that belong to both Set A and Set B. In other words, we are looking for points that lie on the parabola y = x² - 3x + 2 and also on the line y = 0. These are the points where the parabola intersects the x-axis.
We already found the points where the parabola intersects the x-axis when we factored the equation. The solutions to x² - 3x + 2 = 0 are x = 1 and x = 2. Therefore, the intersection of Sets A and B is: A ∩ B = {(1, 0), (2, 0)}. These are the only two points that lie on both the parabola and the x-axis, where both coordinates are integers.
To better understand Sets A and B, it's helpful to visualize them on a coordinate plane. The parabola y = x² - 3x + 2 forms a U-shaped curve that opens upwards. The vertex of the parabola can be found by completing the square or using the formula x = -b / 2a, where a = 1 and b = -3. This gives us x = 3 / 2 = 1.5. The corresponding y-value is y = (1.5)² - 3(1.5) + 2 = 2.25 - 4.5 + 2 = -0.25. So, the vertex is at the point (1.5, -0.25).
The line y = 0 is the x-axis. The points in Set A are discrete points scattered along the parabola, while the points in Set B are all the integer points along the x-axis. The intersection of the two sets are the two points where the parabola crosses the x-axis, (1, 0) and (2, 0).
Set A, the set of integer points on the parabola, has several interesting properties. As we observed earlier, the points in Set A tend to move further away from the x-axis as the absolute value of x increases. This is because the quadratic term x² in the equation dominates the linear term -3x and the constant term 2 when |x| is large. The symmetry of the parabola also means that for any point (x, y) in Set A, there is often another point (3 - x, y) in Set A, reflecting the symmetry around the vertical line x = 1.5, which passes through the vertex.
Set A is an infinite set, as there are infinitely many integer values of x that we can substitute into the equation. However, the points in Set A are discrete, meaning they are isolated from each other. There are gaps between the points, and the density of points decreases as we move away from the vertex.
Set B, the set of integer points on the line y = 0, is much simpler to characterize. It is an infinite set consisting of equally spaced points along the x-axis. The points in Set B are also discrete, but they are evenly distributed. Unlike Set A, there is no symmetry or curvature in Set B, as it is a straight line.
The problem of finding integer solutions to equations is a fundamental topic in number theory and Diophantine equations. Understanding the sets of integer points on curves and lines has applications in various areas of mathematics, including cryptography, computer science, and optimization problems. For instance, finding integer solutions to equations is crucial in designing secure cryptographic systems and solving integer programming problems.
In this article, we have thoroughly explored Sets A and B, which represent integer points on a parabola and a line, respectively. By factoring the quadratic equation and substituting integer values, we identified several elements of Set A. We found that Set B consists of all integer points on the x-axis. The intersection of Sets A and B gave us the x-intercepts of the parabola. Visualizing these sets on a coordinate plane provided a clear understanding of their properties and characteristics. This analysis highlights the interplay between algebra and geometry in solving mathematical problems and emphasizes the significance of integer solutions in various applications.
By examining the equation y = x² - 3x + 2 and the line y = 0, we've demonstrated methods for identifying integer solutions and discussed the properties of the resulting sets. This exploration underscores the richness of mathematical concepts when viewed through the lens of coordinate geometry and number theory.
