BQ #4: Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

Again, we are going to go back to the Unit Circle and as the unit circle tells us, the trig ratio for tan=y/x and cot= x/y since it's the reciprocal of tangent. Another way we can see these ratios is tan=sin/cos and cot= cos/sin.  Look that the paragraphs and the images for more details. With this in mind,  cos would need to equal 0 for tangent and sin would need to equal 0 for cotagent in order to find our asymptotes since they are undefined ratios.

For tangent, Cos would be 0 at 90* (pi/2) and 270* (3pi/2). Before we graph, also remember that the pattern for tangent will be  + - + -. Now, let's label our graph based on the quadrants and their pattern, but highlight where the asymptotes will lie on.
You can see that based on the asymptotes, you need to draw your graph within those two quadrant. It lies on the second quadrant , which is negative and then in quadrant three, which is positive. When you connect those two, your graph will go uphill.

Now as for cotangent, you also need to remember that in order to get an asymptote, sin will need to equal 0. Based on the unit circle, sin=0 at 0* (0pi) and 180* (pi). The pattern for cotangent is the same as it is for tangent ( +-+-). Once we label our graph and highlight the asymptotes, we need to look at the quadrants that lie within those asymptotes. We land on the first quadrant, which is positive and then in the second quadrant, which will be negative. Once we connect these pieces together, we get a downhill graph.