## “I LOVE YOU” Math Graph

This is how to plot “I LOVE YOU” using Math Graphs (many piecewise functions plotted together).

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Source: Found it on Weibo (China’s version of Facebook)

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren’t even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

In Love and Math, renowned mathematician Edward Frenkel reveals a side of math we’ve never seen, suffused with all the beauty and elegance of a work of art. In this heartfelt and passionate book, Frenkel shows that mathematics, far from occupying a specialist niche, goes to the heart of all matter, uniting us across cultures, time, and space.

## 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)

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