Question

Write the rule that defines the function represented by the table

Answer 1

y = 2 * 4^x

Explanation:

We could define a wide variety of rules!

It turns out that any set of n (x, y) pairs can be modeled by a singular polynomial of degree (n-1) or less, and by infinitely many polynomials of degree n or more! And we're only talking polynomials!

Let's input it into a polynomial interpolation calculator:

`(27 x^4)/4 - (63 x^3)/2 + (225 x^2)/4 - (51 x)/2 + 2`

Well I don't love this.

Let's try to actually look for patterns. We see that the function is rapidly increasing. More and more, actually. Let's look for the ratios of each next y

8/2 = 4

32/8 = 4

128/32 = 4

512/128 = 4

BINGO!

So we can define the rule as:

y = 2 * 4^x