## 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 $x=3$.

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 $x=-2$.

## 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 $y=1$.

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 $y=1$.

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 ( $y=ax^2+bx+c$)

If a>0, the graph is a “U” shape or “happy face”. 🙂 If a<0, the graph is a “n” shape or “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 pt: (2,3)

## Graphs of $y=ax^n$

### $\displaystyle y=ax^{-2}=\frac{a}{x^2}$, where a>0 (Volcano shape) Not touching the x-axis (Asymptote) If a<0, the graph becomes upside down. ( $\displaystyle y=\frac{-1}{x^2}$)

### $\displaystyle y=ax^{-1}=\frac{a}{x}$, where a>0. (Hyperbola, Slanted Hourglass)  If a<0, it will be upside down (reflected about x-axis) ( $\displaystyle y=\frac{-1}{x}$)

### Cubic $y=x^3$ 