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

Functions and Graphs

In this post, we will discuss how to sketch the graphs of y=ax^n, for y=-2 to 3.

First, we will look at Quadratic Graphs (y=ax^2+bx+c)

Quadratic Graphs

If a>0, the graph is a “U” shape or “happy face”. 🙂happy face

If a<0, the graph is a “n” shape or “sad face”. 😦

sad face

Intercepts

  • To find x-intercept: Let y=0
  • To find y-intercept: Let x=0

Completing the square

  • E.g. 2x^2+8x+4=2(x^2+4x+2) (take out common factor of x^2)
  • =2(x^2+4x \mathbf{+2^2-2^2}+2) (Key step: Divide coefficient of x by 2, add & subtract the square of it)
  • Check your answer using substitution method! (Sub. in x=9 into your initial and final answer)

Turning Points

y=\pm (x-p)^2+q

* (p,q) is the turning point of the graph

For example, y=(x-2)^2+3

minimum point

Minimum pt: (2,3)

Graphs of y=ax^n

\displaystyle y=ax^{-2}=\frac{a}{x^2}, where a>0 (Volcano shape)

1 over x2
Not touching the x-axis (Asymptote)
volcano

If a<0, the graph becomes upside down.

minus 1 over x2

(\displaystyle y=\frac{-1}{x^2})

\displaystyle y=ax^{-1}=\frac{a}{x}, where a>0. (Hyperbola, Slanted Hourglass)

hyperbolaslanted hourglass

If a<0, it will be upside down (reflected about x-axis)

minus 1 over x(\displaystyle y=\frac{-1}{x})

Cubic y=x^3

cubic