Need an equation to fit this exponential curve

Question

I need a curve that grows exponentially. Only 2 data points are important:

(0, 0)
(1, 1)

After that, I just need to be able to play with how much the graph curves upwards. y=x^2 is the simplest curve that satisfies this, but it jumps up too high for me in the beginning (y=4 when x=2, I would want y to equal something like 2.5) and it doesn't curve up enough as x gets above 8 or so.

Answer 1

You could use $$2.5\cdot 1.4^x−2.5$$

But you could also look at $$5\cdot 1.2^x−5$$

Answer 2

What about $y = 2^x/2$? The first five coordinates are {{0,0),(2,2),(3,4),(4,8)}.? You can't have an exponential function just "not curve up" until 8 or so.