Sketch the graph on the axis y=e^2x+1 and y=e^x

Question

Sketch the graph on the axis y=e^2x+1 and y=e^x

 

Answer 1

The graph of y=e^x has y-intercept at (0,1), and horizontal asymptote the line y=0 (i.e. the x-axis).  

 

Two other points on the graph of y=e^x are (-1,1/e) and (1,e)

 

 

To plot the graph of y=e^2x+1, we are simultaneously:

 

Stretching the graph to be twice as tall as e^x, as the value for x need only be half as much as before (see the exponent of e - i.e. "2x"), in order to attain the previous corresponding values for y, and:

 

Shifting the graph vertically up one by adding one (+1) to the exponential function.  This has a net effect of not only shifting all points on the graph vertically upwards (in which case the y-intercept is now at (0,2)), but it also shifts the horizontal asymptote up one unit as well - to the line y=1.