graph the function. (a graphing calculator is recommended. select the correct graph.) ( f(x)=3^{x + 1} )

Explanation:

Step1: Analyze the exponential - type function

The function is (f(x)=3^{x + 1}). We know that the general form of an exponential function is (y = a\cdot b^{x - h}+k), where in our case (a = 3), (b = 3), (h=-1), and (k = 0).

Step2: Find the y - intercept

To find the y - intercept, set (x = 0). Then (f(0)=3^{0 + 1}=3^{1}=3).

Step3: Analyze the behavior as (x\to-\infty)

As (x\to-\infty), (3^{x + 1}=\frac{3}{3^{-x}}\to0). The function approaches the x - axis ((y = 0)) from above as (x\to-\infty).

Step4: Analyze the behavior as (x\to+\infty)

As (x\to+\infty), (3^{x + 1}\to+\infty) since the base (b = 3>1).

The graph of (y = 3^{x + 1}) is an exponential growth function that passes through the point ((0,3)) and approaches the x - axis as (x\to-\infty) and goes to (+\infty) as (x\to+\infty).

Answer:

The graph that has a y - intercept at (y = 3), approaches the x - axis as (x\to-\infty) and increases without bound as (x\to+\infty) is the correct one. (Since no specific labels for the graphs are given in the problem - statement, you would identify the graph with these characteristics among the options provided).