graph the function.\ny = e^x + 1

Explanation:

Step1: Find the y - intercept

Set (x = 0). Then (y=e^{0}+1=1 + 1=2).

Step2: Analyze the asymptote

As (x\to-\infty), (e^{x}\to0), so (y = e^{x}+1\to1). The horizontal asymptote is (y = 1).

Step3: Analyze the behavior for positive x

As (x\to+\infty), (e^{x}\to+\infty), so (y=e^{x}+1\to+\infty). The function is an increasing - function since the derivative of (y = e^{x}+1) is (y'=e^{x}>0) for all real (x).

To graph the function, plot the y - intercept at the point ((0,2)), draw the horizontal asymptote (y = 1) as a dashed line, and sketch an increasing curve that approaches the asymptote as (x\to-\infty) and goes to (+\infty) as (x\to+\infty).

Answer:

Graph with y - intercept at ((0,2)), horizontal asymptote (y = 1), and an increasing curve.