Why is a "normal" tangent graph uphill, but a "normal" tangent graph downhill?
If you take a look at the graph of tangent (top), the asymptotes starting from (0,0) are at pi and 3pi/2. These are also where cosine is zero and touches the x-axis. When remembering trig identities, tan = sin/cos. If Cosine is zero, that means tan is undefined and it's graph cannot exist where cosine is zero which is why the asymptotes are there. It goes uphill because starting from (0,0) the first quadrant it 0 to pi/2 where tan is positive. The graph goes up and along the asymptote positively. After the asymptote, tan is negative II quadrant but positive III quadrant so the graph starts negative up against (but not touching!) the asymptote and then curving up positive against the asymptote making an uphill curve.
For the cot graph (bottom), the asymptotes are shifted! The asymptotes are now at 0 and pi, where sine is 0. The identity states that cot = cos/sin, the reciprocal of tan's identity. Where cosine is zero, cot is undefined and the cot graph cannot touch where cosine is zero. The graph goes downhill because the asymptote starts at (0,0) and quadrant I starts there. Thus the graph will start against the asymptote going down into quadrant II which is negative and staying there because of the next asymptote at pi. This makes the downhill curve.