#power function

Domain & Range

Graphing Example of Multitude

Unit 20: Dividing Polynomials

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Long Division Basics

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In this post I will show you how to create an ease in/out function using power functions, i.e., functions of the form f(x)=cxa. While c and a in a power function could be any real number, here we will omit the c variable (c=1) and use values greater than 1 for a. Let's see how a power function looks like.

Every power function has a very usefull characteristic; f(1)=c because 1a=1 for all a. In our case this means that f(1)=1 as we set c=1 above. If we bound the domain of the function to the closed interval [0,1] then we have a curve that always begins at (0,0) and ends at (1,1) regardless of the a value. This curve can be scaled very easily in both dimensions to fit our needs.

The above curve (in red) could be used as an ease in function, but what if we also want a smooth ending to our transition? To achieve this we will use a second power function and some basic school math. Actually we will start with a same function and we will try to flip and shift it in order to dock at the end of the first one.

f(x)=xa, g(x)=xa

Step 1 : flip vertically

g(x)=xa → g(x)=-xa

Step 2 : flip horizontally

g(x)=-xa → g(x)=-(-x)a

* Notice that -(-x)a is different than - -(x)a, so you can't omit the minus symbols!

* Programmatically step 1 and step 2 can be replaced with a single condition check. If a is odd then we should begin directly from step 3. If a is even we must flip g vertically (step 1) and then continue to step 3 (omitting step 2). This is because power functions with odd exponent have different type of symmetry than those with even exponent.

Step 3 : shift vertically

g(x)=-(-x)a → g(x)=-(-x)a+2

Step 4 : shift horizontally

g(x)=-(-x)a+2 → g(x)=-(-x+2)a+2

We could stop in this step but our function starts at (0,0) and ends at (2,2). It would be more convenient if our function ended at (1,1).

Step 5 : normalize

f(x)=xa → f(x)=(x*2)a/2

g(x)=-(-x+2)a+2 → g(x)=(-(-x*2+2)a+2)/2

Step 6 : combine

f(x)=(x*2)a/2 for 0 ≤ x ≤ 0.5 and f(x)=g(x)=(-(-x*2+2)a+2)/2 for 0.5 ≤ x ≤ 1

With this formula you can produce a variety of curves by just adjusting the value of a.