How to build a custom tan function that goes through specified points Intro
I often need trigonometric functions while programming, but I'm not always able to get what I want — especially when dealing with $tan$. Every time I need it, I spend hours trying to figure out the function.
Request
What I really need is a function like the one in this graph, where 0 < x < 1 and fn(0.5) = 0.5, with a variable slope (curves more or less accentuated)

Current work
I was able to plot that on Grapher.app with this equation, but it's manually tweaked to hit those points: $y={\tan(2.78x-1.38)\over\pi^2}+0.5$
Every time I need to change the slope, that function requires infinite tweaking to make the ends match [0,0] and [1,1]
Notes
It doesn't have to be a $tan$, just anything that resembles that graph. Even pointing me to anything that would help me understand how to warp trig functions to touch my points would be greatly appreciated.
 A: Look at $y=\tanh mx / \tanh m$ for starters. Then find the inverse function to get the shape right. After that, a linear change of coordinates should do it. You see?
This gives $y = \frac12[1+\frac1m \tanh^{-1}((2x-1)\tanh m)]$. Varying $m$ will change the slope at the center point $(1/2,1/2)$.
A: The general function is of the form:
$$f(x) = A\tan(B(x-C))+D$$
The "center point": This function goes through the center point $(C, D)$. In your case you want $C=\dfrac{1}{2}, D=\dfrac{1}{2}$.
The "diagonal points": Tangent is an odd function, meaning this "shifted" version has the property that if $t>0$, $f(C+t) = -f(C-t)$. The values of these depend on the value of $A$, the "vertical stretchyness" of the graph. If you plug in $x=\dfrac{\pi}{4B}+C$ then $f(x) = A+D$. Since you want to go through $(1,1)$ you need 
$$A+D=1 \ \ \text{  and  } \ \ \dfrac{\pi}{4B}+C=1$$
Since $C,D=\frac{1}{2}$ this means $B$ must be $B=\frac{\pi}{2}$ and $A=\dfrac{1}{2}$. By symmetry, it will also pass through $(0,0)$.
$$f(x) = \dfrac{1}{2}\tan\left(\dfrac{\pi}{2}\left(x-\dfrac{1}{2}\right)\right) + \dfrac{1}{2}$$
is the desired function for your above graph.
