Basically, a normal tangent graph is uphill because it is based on its asymptotes. They are located at pi/2 and 3pi/2. Furthermore, the quadrants are shown as positive, negative, positive, and negative (based on sin/cos). Cotangent is downhill because its asymptotes are located at 0 and pi. Because of where the asymptotes are located, it is downhill. The quadrants are positive, negative, positive, negative. The asymptotes are where sine and cosine are equal to 0 (making the ratio undefined).
Visual of Tangent
As you can see, the colors represent the different quadrants (I,II,III,IV)
Visual of Cotangent
Asymptotes at 0 radians and pi