## 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

(mouth)

(yellow region)

(left eye)

(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:

- Heart: An equation that generates a beautiful or unique shape for motivating students in mathematics – Output in Wolfram Alpha
- Slanting heart: LOVE +MATH = can you read this formula? – Output in Geogebra
- …and much more!

### Some learning

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

**Change of origin:**For example, is a circle, centred at (0,0) and radius 0. While, 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.**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 , with origin moved from (0,0) to (0,-3) (and equation becomes ) and “*made wider”*by multiplying it with a scalar, in this case, , that is, equation becomes . If you notice the slanting heart example above, changing the value of b makes the graph bigger/smaller accordingly.**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.**Limiting Domain:**Another trick is to limit the domain of definition. In Geogebra, it can be achieved in two different ways:**“**Function(expression, start value, end value)*Function*” function:*Example:*Function((x^2-3)*1/2, -1, 1) plots the expression (x^2-3)*1/2 in the domain [-1,1].**“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

To start with, try “drawing” Olympic Rings.

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

### Challenge** 3**

Pick another simple image, and try plotting it!

## Step 3: Do some Dance!

Post your moves in the comments!