.
It is easier if we graph y=2^x as the base function first. This is an exponential function. To do this, we pick a few values for x and plug them into the function to find the corresponding y for each one:
x=1, y=2^1=2
x=2, y=2^2=4
x=3, y=2^3=8
x=oo, y=2^oo=oo
x=-1, y=2^-1=1/2^1=1/2
x=-2, y=2^-2=1/2^2=1/4
x=-3, y=2^-3=1/2^3=1/8
x=-oo, y=2^-oo=1/2^oo=1/oo=0
These results indicate that as x goes to oo, y goes to oo, and as x goes to -oo, y goes to 0, which means the y-axis is the horizontal asymptote. Plotting the points on a coordinate system gives us the graph:
graph{2^x [-10, 10, -5, 5]}
We know that if we add a constant to our x the graph shifts in the negative direction by that number of units.
We also know that if we subtract a constant from our y the graph will shift down by that number of units.
Therefore, if we add 2 to our x and subtract 1 from our y the graph will shift 2 units to the left and 1 unit down, and our function becomes:
y=2^(x+2)-1
The graph of this function is:
graph{2^(x+2)-1 [-10, 10, -5, 5]}