Question

What is the range of y=2^x+1?

Answer 1

You could draw up a table and calculate the y-values for a set of x-values, like this:

x -4 -3 -2 -1 0 1 2 3 4
y 16 9 4 1 0 1 4 9 16
This gives us a series of points (-4,16), (-3,9), (-2,4) up to (4,16).

You then join these dots with a smooth curve and get something like the following.



Notice that the vertex of the parabola (the “pointy” end) is at the origin, (0, 0).

Now for all the curves that I draw below, I’m not going to draw up a table. It becomes tedious, and it can lead to incorrect graphs. It is better to be able to recognize the graph type (from the equation) and then know how to sketch it in the right place and with the right orientation.

I will consider the effect of small changes to the equation and then sketch my curve.

All of the following graphs have the same size and shape as the above curve. I am just moving that curve around to show you how it works.

Answer 2

all real but not zero

Answer 3

and 1 also