For problem 1(a) equation 8: $u_t + (x^2+1)u_x = 0$

From solving for the characteristic equation I get $x = \tan(t-k)$ where k is a constant.

My question is how would I draw the characteristics in this case? The periodic nature of tan would result in just an indistinguishable blob of lines if I were to plot it out? Should I just draw the lines for one value of k?