Why is a negative exponent present in Fourier and Laplace transform?

could anyone explain why there is a need of negative exponent in fourier and laplace transform.I looked through the web but I couldn't get anything.Does anything happen if a positive exponent is placed in these transforms.

While looking through http://1drv.ms/1tbV45S it says that if $s>0$ it becomes a rapidly decreasing function while if $s<0$ it becomes an rapidly increasing functin of t.I couldn't understand that.Can anyone illustrate this.