Question

graph the equation below using a calculator and point - by - point plotting. indicate increasing and decreasing intervals. y = 2 ln(x + 8)
choose the correct graph below.
where is the graph increasing and/or decreasing? select the correct choice below and fill in any answer boxes within your choice.
a. the graph increases on and decreases on.
b. the graph increases on but does not decrease.
c. the graph decreases on but does not increase.
d. the graph neither increases nor decreases.

Explanation:

Step1: Determine domain

The argument of the natural - logarithm function must be positive. So, \(x + 8>0\), which gives \(x>-8\). The domain of \(y = 2\ln(x + 8)\) is \((-8,\infty)\).

Step2: Analyze the derivative

The derivative of \(y=f(x)=2\ln(x + 8)\) using the chain - rule. The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{u'}{u}\), where \(u=x + 8\) and \(u'=1\). So \(y'=\frac{2}{x + 8}\).
For \(x>-8\), \(y'=\frac{2}{x + 8}>0\). When the derivative of a function is positive on an interval, the function is increasing on that interval.

Answer:

The graph increases on \((-8,\infty)\) but does not decrease. So the answer is B. The graph increases on \((-8,\infty)\) but does not decrease.