MTH403 Lectures 23-28

1. Introduction

Graphs are beneficial because they summarize and display information in a manner that is easy for most people to comprehend. Graphs are used in many academic disciplines, including math, hard sciences and social sciences. They make appearances in corporate settings, serving as useful tools to convey financial information and facilitate data analysis. Different graphs are used depending on the information that individuals wish to convey. Many graphs are used to concisely and clearly summarize data; the best type of graph to use depends on the type of data being conveyed (such as nominal, scale-discrete, scalecontinuous and ordinal). Data summary graphs are generally nominal or contain data that can be reduced in some way; pie charts and bar charts are common and popular examples. In the coming lectures, we discuss the graphs in a rectangular coordinate system. Equations can be graphed on a set of coordinate axes. The location of every point on a graph can be determined by two coordinates, written as an ordered pair, (x, y). These are also known as Cartesian coordinates, after the French mathematician Rene Descartes, who is credited with their invention. Although the use of rectangular coordinates in such geometric applications as surveying and planning has been practiced since ancient times, it was not until the seventeenth century that geometry and algebra were joined to form the branch of mathematics called analytic geometry. French mathematician and philosopher Rene Descartes (1596-1650) devised a simple plan whereby two number lines were intersected at right angles with the position of a point in a plane determined by its distance from each of the lines. This system is called the rectangular coordinate system (or Cartesian coordinate system).

2. Outlines

• The Coordinate Plane.
• Asymptotes.

2. 1. The Coordinate Plane. The coordinate plane is a plane determined by two perpen- dicular lines, the x-axis and the y-axis. The x-axis is the horizontal axis, and the y-axis is the vertical axis. Every point in the plane can be stated by a pair of coordinates that express the location of the point in terms of the two axes. The intersection of the x- and y-axes is designated as the origin, and its point is (0, 0). As you can see from the figure, each of the points on the coordinate plane is expressed by a pair of coordinates: (x, y). The first coordinate in a coordinate pair is called the x- coordinate. The x-coordinate is the point’s location along the x-axis and can be determined by the point’s distance from the y-axis (where x = 0). If the point is to the right of the y-axis, its x-coordinate is positive, and if the point is to the left of the y-axis, its x-coordinate is negative. The second coordinate in a coordinate pair is the y-coordinate. The y-coordinate of a point is its location along the y-axis and can be calculated as the distance from that point to the x-axis. If the point is above the x-axis, its y-coordinate is positive, and if the point is below the x-axis, its y-coordinate is negative.

Definition 2.1. An asymptote is a line to which the graph gets arbitrarily close. That means that for any distance named, no matter how small, the graph will get within that distance and stay within that distance for some section of the graph with infinite length. More precisely

Definition 2.2. The graph of y = f (x) has a horizontal asymptote of y = b, if and only if, either

lim x→∞ f (x) = b or lim x→−∞ f (x) = b. (2.1)

As x →±∞, the curve approaches some constant value b.

Definition 2.3. The graph of y = f (x) has a vertical asymptote of x = c, if and only if, either

lim x→c− f (x) = ±∞ or lim x→c+ f (x) = ±∞. (2.2)

As x →±c, then the curve goes towards ±∞. Definition 2.4. Let r(x) be a rational function with polynomial p(x) = anxn + ... + a0 of degree n in the numerator and polynomial q(x) = bmxm + ... + b0 of degree m in the denominator

• If n < m, then r(x) has a horizontal asymptote of y = 0.
• If n > m, then r(x) becomes unbounded for large values of x (positive or negative).
• If n = m, then r(x) has a horizontal asymptote of y = an bn .

r(x) = 1 x , (2.3)

this function is defined ∀x̸ = 0. Consider the sequence of numbers xn = 1 2 , 1 3 , 1 4 , 1 5 , · · · , 1 k for k = 1, 2, · · · , n. These numbers are getting closer and closer to zero. Since r(xn) = xn = n = n. So, r(xn) = 2, 3, 4, · · · , k, · · · for n = 2, 3, 3, · · · , k, · · · , which is getting larger and larger, so approaching the vertical line x = o. Thus there is a vertical asymptote at x = 0.