Whenever I am asked to find a formula from a range of values, I always use difference patterns.

Do with the data given:

x____________1_______2________3________4

y____________3_______5________7________9

$1^{st}$ diff:___________+2______+2________+2

I know that with the general formula $y = mx + c$ the difference pattern is:

x____________1_______2________3________4

y__________m + c____2m + c____3m + c_____4m + c

$1^{st}$ diff:_________+m______+m _______+m

So the first difference pattern is equal to $m$, i.e. $m = 2$.

I can then use $y(1) = m + c$ to find $c$

where $y(1) = 3$, and $m = 2$

$3 = 2 + c$

therefore $c = 1$

and putting it all together we get $y = 2x + 1$

You can use further difference patterns for polynomials such as $2^{nd}$ diff for $x^2$, $3^{rd}$ diff for $x^3$, etc. but note that for an exponential function you will need to look at a common ratio, rather than a difference pattern.