Question

Sketch the following sets of points in the \(x-y\) plane. $$ \left\\{\left(x, \frac{x^{2}}{y}\right): x \in \mathbb{R}, y \in \mathbb{N}\right\\} $$

Short answer

The sketch of the set of points \(\left\{\left(x, \frac{x^{2}}{y}\right): x \in \mathbb{R}, y \in \mathbb{N}\right\}\) in the \(x-y\) plane generates a series of parabolas, with vertices at \(x = 0\) and oriented upwards for each \(y\) value in natural numbers. As we increase the value of \(y\), the parabolas become compressed vertically. This pattern continues for both positive and negative real \(x\) values, since the graph is symmetric across the y-axis.

Step-by-step solution

Identify The Form of The Function

The function given, \(f(x,y) = \left(x, \frac{x^{2}}{y}\right)\), is a parametric function where \(x\) is the parameter. The \(x\) values of the function will take any real number. The \(y\) values are the result of the operation \(\frac{x^{2}}{y}\), where \(y\) is any natural number. Notice that we have two domains to satisfy: \(x \in \mathbb{R}\) and \(y \in \mathbb{N}\). Therefore, varying the values of \(x\) and \(y\) accordingly to these sets we can draw our set of points.

Sketch By Varying \(x\) and \(y\)

Start by fixing a value for \(y\) and vary \(x\) across all real values. Then repeat this process for each natural number \(y\). For instance, when \(y = 1\), the function transforms into \(f(x,y) = \left(x, x^{2}\right)\), which is a standard parabola. By changing the \(y\) value to \(2, 3, 4, ...\), we get different functions i.e. \(f(x,y) = \left(x, \frac{x^{2}}{2}\right)\), \(f(x,y) = \left(x, \frac{x^{2}}{3}\right)\), and \(f(x,y) = \left(x, \frac{x^{2}}{4}\right)\) respectively, which are parabolas but stretched or compressed vertically according to the \(y\) value. Remember, since \(y\) are natural numbers, we have separate points for each \(y\), not interconnecting lines. Plot the points on the \(x-y\) plane.

Repeat for Negative Real Values of \(x\)

The process should also be repeated for negative values of \(x\), since \(x\) can take any real number. Notice that the function will be symmetric across the y-axis, because for any negative \(x\), squared operation will give a positive output, hence the result will still be positive. So repeat the process like in the previous step, but this time with negative \(x\) values, and plot the points on the \(x-y\) plane.

Key concepts

Coordinate Plane

Imagine the coordinate plane, often referred to as the Cartesian plane, as a two-dimensional surface on which we can plot points, lines, and curves. It is made up of two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), that intersect at a point called the origin (0,0). Each point on this plane is identified by an ordered pair \( (x, y) \) in which \( x \) corresponds to a position along the x-axis and \( y \) along the y-axis.

In the exercise, you are tasked to plot points generated by a set of parametric equations on this plane. The coordinate plane allows us to visually represent the relationship between the varying \( x \) and \( y \) values generated by these equations. By plotting multiple points, we can begin to see the shape, or the 'sketch', of the equation as a curve or a series of points on the plane.

Real Numbers

The real numbers \( \mathbb{R} \) include all the numbers on the number line. They cover all the possible values without gaps or jumps, from the infinitely small to the infinitely large, including all fractions and decimals. The real numbers are essential for measuring continuous quantities and are used in the exercise to represent the domain of the \( x \) values in the parametric equations.

In the exercise, you are instructed that \( x \) takes on values from the real numbers. This means that \( x \) can be any value: positive, negative, zero, rational (like \( \frac{1}{2} \) or \( 4.5 \) ), or even irrational (like \( \sqrt{2} \) or \( \pi \) ). Understanding the vastness of the real number domain is critical for correctly varying \( x \) to sketch the set of points accurately.

Natural Numbers

The natural numbers \( \mathbb{N} \) are the counting numbers starting from 1, and moving upwards—1, 2, 3, and so on. They are the most basic set of numbers used for ordering and quantification in mathematics. Unlike the real numbers, the natural numbers are discrete, meaning they are countable and there are gaps between them as they go from one integer to the next. They do not include zero, fractions, decimals, or negative numbers.

In relation to the exercise, the values that \( y \) can take are restricted to natural numbers. This limitation ensures that each resulting point from the parameterized equations forms a distinct and separate location, which is important for the correct plotting of the function on the coordinate plane.

Parametric Equations

Parametric equations represent a set of related quantities as functions of one or more independent variables, known as parameters. These equations describe the coordinates of the points making up a geometric shape not by expressing \( y \) directly as a function of \( x \) but rather by linking \( x \) and \( y \) separately to a third variable, often \( t \) or in our case, \( x \) itself as the independent parameter and \( y \) as the dependent natural number. The beauty of parametric equations lies in their ability to express complex curves and shapes that might be difficult or impossible to describe with a single equation in \( x \) and \( y \) alone.

In this exercise, the parametric equations given produce a set of points in the \( x-y \) plane. By manipulating the parameter \( x \) over the real numbers and \( y \) over the natural numbers, one can sketch a variety of points which reflect the combined behavior of these two varying quantities within the constraints of the respective domains. The sketch would showcase how the shape changes for different values of \( y \) over a continuous spectrum of \( x \) values.