How do you graph #y = -tan(x-pi) -3#?

1 Answer
Oct 21, 2015

Answer:

Shift the graph of #tan(x)# to the right of #\pi# units, then reflect it with respect to the #x#-axis, then shift it downwards of #3# units.

Explanation:

Assuming you know the graph of #tan(x)#, let's analise step by step what changes you make, and their consequences.

First step: from #tan(x)# to #tan(x-\pi)#. This changes is one of the form #f(x)->f(x+k)#. This kind of changes means a horizontal translation of #k# units, to the left if #k# is positive, to the right if #k# is negative.

So, in you case, we start from the graph of #tan(x)#, and shift it to the right of #\pi# units.

Second step: from #tan(x-\pi)# to #-tan(x-pi)#. This is one of the most simple and intuitive changes: if you go from #f(x)# to #-f(x)#, you simply change the sign of every single value of the function, resulting in a reflection with respect to the #x#-axis.

Third step: from #-tan(x-pi)# to #-tan(x-pi)-3#: this is a change of the form #f(x)->f(x)+k#. This means that you need to add a certain number to the values of the function, resulting in a vertical shift, which will be upwards if #k# is positive, and downwards if #k# is negative. So, in your case, you need to shift the graph of #-tan(x-pi)# down of three units.