How to create a cubic spline between the lines x=0 and y=1?

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I am trying to create a simple cubic spline from points (0,0) to (m,1) connecting the lines y=1 and x=0.

However, I am having trouble getting the spline to be tangential to the x=0 line at (0,0).

https://www.desmos.com/calculator/gbyugajoyi

I presume this means I can't provide the derivative of x=0 to get it to do what I want?

How can I fix this?

I tried using y = vx where "v" is an arbitrarily high constant instead of x=0, and it works, but only at very low "m" values. At higher "m", it switches the spline around to connect the wrong way.

https://www.desmos.com/calculator/ijyna1uwtf

Is that controllable or fixable either?