Question

I need help with this practice since it’s from my act prep guide it asks NOT to use a graphing website or tool to answer this, so please, if you can, answer it either on a drawing tool or pen and paper

Answer 1

The function is given to be:

`f(x)=\cot (x+(\pi)/(6))`

A full period of a tan graph is π. Hence, we will make sure our table of values will cover more than π units for the x-axis.

To get the first value we can use in the table of values, we will equate the function to 0 and solve for x:

`\begin{gathered} \cot (x+(\pi)/(6))=0 \\ x+(\pi)/(6)=\cot ^(-1)0 \\ x+(\pi)/(6)=(\pi)/(2) \\ \therefore \\ x=(\pi)/(2)-(\pi)/(6) \\ x=(\pi)/(3) \end{gathered}`

We will take intervals of π/6, such that we will use values of x to be:

`\begin{gathered} x_1=(\pi)/(3)+(\pi)/(6)=(\pi)/(2) \\ x_2=(\pi)/(2)+(\pi)/(6)=(2\pi)/(3) \\ x_3=(2\pi)/(3)+(\pi)/(6)=(5\pi)/(6) \\ x_4=(5\pi)/(6)+(\pi)/(6)=\pi \\ x_5=\pi+(\pi)/(6)=(7\pi)/(6) \\ x_6=(7\pi)/(6)+(\pi)/(6)=(4\pi)/(3) \end{gathered}`

If we substitute these values into the function, we can prepare a table as shown below:

Note that there is a vertical asymptote at x = 5π/6.

Using this table, the graph is drawn as shown below: