# Equation for a smooth step function

http://i.stack.imgur.com/t1PlV.jpg

The equation must be smooth and satisfy the following two conditions:

• When $x \leq -10, y=-50$
• When $x \geq 10, y=50$

Is there an equation (or multiple equations) that lets me graph a curve that looks like that?

The sigmoid function, $$S(x) = \frac{1}{1+e^{-x}}$$, achieves close to what you need (with appropriate scaling and shifting of the function).
Do you need the function to be exactly $$\pm 50$$ when evaluated at $$\pm 10$$? If so, a polynomial option would be to use something called the smoothstep function, built using the function $$f(x) = 3x^2-2x^3$$. Modified to your problem, the desired function, denoted by $$g(x)$$, is $$g(x) = \begin{cases} -50 & \text{ if } x\le -10\\ 100\left(3(0.05x+0.5)^2-2(0.05x+0.5)^3 - 0.5\right) & \text{ if } -1010 \end{cases}$$
$$f(x) = \begin{cases} -50 & \quad x<-10\\ 50\sin(\pi x/20) & \quad -10 \le x \le 10 \\ 50 & \quad x >10 \end{cases}$$
There exist smooth complactly supported functions on $\mathbb{R}$, namely bump functions. Pick any such function and integrate it to obtain a monotonic smooth function $F$ that is zero for small enough values and is some non-zero constant for large enough values. Then you can obtain a graph like on the picture with an appropriate affine transformation. ($aF(bx+c)+d$ for some $a,b,c,d\in\mathbb{R}$)