Tag: asymptote

How to Find Asymptotes of Graphs

This post is all about finding  Vertical and Horizontal asymptotes of graphs.

Vertical Asymptotes

Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, \displaystyle y=\frac{2}{x-3}. When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there.

Hence, to find the asymptote, let the denominator be 0. E.g. x-3=0, so x=3.

graph1
\displaystyle y=\frac{2}{x-3}

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. y=\ln (x+2).

\ln 0 is undefined, so when x+2=0 or x=-2, there will be a vertical asymptote at x=-2.

graph2
y=\ln (x+2)

 

Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as x approaches infinity.

To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number).

For example, \displaystyle y=\frac{1}{x}+1. When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is y=1.

graph3
\displaystyle y=\frac{1}{x}+1

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, y=e^{-x}+1. When x is very large, e^{-x} will be very small, and hence y approaches 1. This means that the Horizontal Asymptote will be y=1.

exponential
y=e^{-x}+1

Note: The graphs above were drawn using the software Geogebra. 🙂

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra

Interactive Astronomy using 3D Computer Graphics

This is a school project on using 3D Computer Graphics. It explores a phenomenon whereby a sundial can actually go backwards in the tropics!

Abstract:

Understanding spherical astronomy requires good spatial visualization. Unfortunately, it is very hard to make good three dimensional (3D) illustrations and many illustrations from standard textbooks are in fact incorrect. There are many programs that can be used to create illustrations, but in this report we have focused on TEX-friendly, free programs. We have compared MetaPost, PSTricks, Asymptote and Sketch by creating a series of illustrations related to the problem of why sundials sometimes go backwards in the tropics.

sphere

In it, the Hezekiah Phenomenon is being discussed.

Quote: First, we would like to explain where the name Hezekiah Phenomenon comes from. In the Bible there is a story about God making the shadow of the sundial move backward as a sign for King Hezekiah.

The Bible gives two versions of the story of King Hezekiah and the sundial. First in 2 Kings, Chapter 20.

8 And Hezekiah said unto Isaiah, What [shall be] the sign that the LORD will heal me, and that I shall go up into the house of the LORD the third day? 9 And Isaiah said, This sign shalt thou have of the LORD, that the LORD will do the thing that he hath spoken: shall the shadow go forward ten degrees, or go back ten degrees? 10 And Hezekiah answered, It is a light thing for the shadow to go down ten degrees: nay, but let the shadow return backward ten degrees. 11 And Isaiah the prophet cried unto the LORD: and he brought the shadow ten degrees backward, by which it had gone down in the dial of Ahaz. (2 Kings 20: 8–11, King James Version)