graph y = ln (x + 6).

Explanation:

Step1: Identify the domain

The argument of the natural - logarithm function must be positive. So, (x + 6>0), which gives (x>-6). The domain is ((-6,\infty)).

Step2: Find the vertical asymptote

As (x\to - 6^{+}), (y = \ln(x + 6)\to-\infty). So, the vertical asymptote is (x=-6).

Step3: Find the (x) - intercept

Set (y = 0). Then (\ln(x + 6)=0). Since (\ln1 = 0), we have (x+6 = 1), so (x=-5). The (x) - intercept is ((-5,0)).

Step4: Analyze the behavior

The derivative of (y=\ln(x + 6)) is (y'=\frac{1}{x + 6}). Since (y'>0) for (x>-6), the function is increasing on its domain ((-6,\infty)).

To graph the function, draw a curve that approaches the vertical asymptote (x = - 6) from the right, passes through the point ((-5,0)), and increases as (x) increases.

Answer:

Graph a curve with vertical asymptote (x=-6), passing through ((-5,0)) and increasing for (x > - 6).