Graphing Exponential Functions With e, Transformations, Domain and Range, Asymptotes, Precalculus

The video starts by introducing how to graph exponential functions using a simple example like y = 2^x. It explains the importance of making a table with points like 0 and 1, and identifying the horizontal asymptote, which for this basic function is y = 0. It further demonstrates how to plot these points to sketch the graph and determine its domain (negative infinity to infinity) and range (0 to infinity, not inclusive of 0).

This section explains how to graph exponential functions with transformations, such as y = 3^(x+1) - 2. It shows how the '-2' shifts the horizontal asymptote down to y = -2 and how '(x+1)' shifts the graph one unit to the left. The video details how to adjust the x-values for plotting to account for these shifts and then determines the domain and range (negative infinity to infinity for domain; -2 to infinity for range).

The video then moves on to graphing functions involving the base 'e', like y = e^x - 1. It clarifies that 'e' is simply a number (approx. 2.7) and the graphing rules remain the same. It demonstrates how to calculate y-values for x=0 and x=1, identify the horizontal asymptote (y = -1), plot the points, and sketch the graph. The section also covers finding the domain (negative infinity to infinity) and range (-1 to infinity).

Finally, the video covers more complex transformations, including reflection, using the example y = 3 - e^(x-2). It advises setting the exponent to 0 and 1 to find appropriate x-values for plotting (in this case, 2 and 3). It highlights that the constant '3' determines the horizontal asymptote (y = 3) and the negative sign in front of 'e' indicates a reflection over the horizontal asymptote. The video explains how this reflection changes the range from being above the asymptote to below it, resulting in a range of (negative infinity to 3).