# Puzzle: Do the Dance!

## Step 1: Finding Nemo

Can you find the function that is given in the graph below?

## Step 2: Drawing Pictures with Functions

Functions are fun.

Let’s take step 1 to a new level. Imagine the smiley face. Can we, rewrite it into a couple of equations? For example,

was plotted with equations, given by

$f(x)= \frac{1}{2}(x^2-3), -1.6\le x\le 1.6,$ (mouth)

$x^2+y^2\le 5$ (yellow region)

$(x+1)^2+(y-1)^2\le 0.2$ (left eye)

$(x-1)^2+(y-1)^2\le 0.2$ (right eye)

Here’s the Geogebra file which plots the equations above.

### Challenge 1

Can you tweak the equations above (Specifically the first one) to obtain the following?

This idea is not a new one – here’s a more complicated image (“Batman Equation”) discussed in math.stackexchange.com. You can see it in action in Wolfram Alpha. Here are some popular shapes that might interest you:

### Some learning

Here’s some essential information that you might require to get started:

1. Change of origin: For example, $x^2+y^2=a^2$ is a circle, centred at (0,0) and radius 0. While, $(x-1)^2+(y-1)^2=a^2$ is a circle, centred at (1,1) and radius a. In other words, we have moved the origin of the circle from (0,0) to (1,1) by tweaking the equation accordingly.
2. Scaling: We can scale the equation by dividing or multiplying it with a scalar. For example, the mouth in the smiley above is essentially the curve $y=x^2$, with origin moved from (0,0) to (0,-3) (and equation becomes $(x^2-3)$) and “made wider” by multiplying it with a scalar, in this case, $\frac{1}{2}$, that is, equation becomes $\frac{1}{2}(x^2-3)$. If you notice the slanting heart example above, changing the value of b makes the graph bigger/smaller accordingly.
3. Flip/mirror: You can flip/mirror a function by tweaking it as well. For example, notice the slanting heart example above, and try to see how the graph changes for b=1 and b=-1.
4. Limiting Domain: Another trick is to limit the domain of definition. In Geogebra, it can be achieved in two different ways:
1. Function” function:Function(expression, start value, end value)
Example: Function((x^2-3)*1/2, -1, 1) plots the expression (x^2-3)*1/2 in the domain [-1,1].
2. “If” function: If(range, expression)
Example: If(-1<=x<=1,(x^2-3)*1/2) produces the same result as above.

### Why should you try this?

If you are an artist and wondering how math comes into the picture in art, this is a good start!

Mathematically, it helps you learn how to manipulate functions better. You can also gain a solid understanding of functions, learn a good set of examples for functions, and help you understand concepts related to functions like continuity, discontinuity, etc. better.

### Challenge 2

It involves plotting five circles, and “changing the origin”.

### Challenge 3

Pick another simple image, and try plotting it!