Start from the parent exponential function y=ab^x. What is its y-intercept? What is the equation for its asymptote? What happens if you translate it vertically?
The graph of a function of this form passes through the point (0,a) and has an asymptote at the line y=0. If we add a constant k to the function the graph gets shifted vertically, and so does the asymptote.
As we can see from the graph above, for our asymptote to be y=2 we can choose k=2. Then, since the y-intercept is given by a+k and we want it to be 5, we have the condition 5=a+k. From here we can solve for a.
So far we have the exponential function y= 3b^x+2. We can now choose any positive real number for the base. For example, b=8. It does not matter which one we choose, since any positive number raised to the zeroth power is 1. We have found the equation y= 3*8^x+2. We can graph it to verify that it satisfies the exercise's conditions.
As we can see, the function has a y-intercept of 5 and an asymptote at y=2, as required. Notice that this is only one example solution, since we could have chosen any positive base and it would work as well. Therefore, there are infinitely many solutions satisfying the exercise's requirements.
