Start with the parent function y = tan(x)
Plot the vertical asymptotes at x = π/2 + kπ, where k is any integer
Plot the x-intercepts at x = kπ, where k is any integer
Draw the repeating increasing curve between each pair of asymptotes
Make the curve pass through points like (-π/4, -1), (0, 0), and (π/4, 1)
Repeat the pattern every π units
For y = a tan(b(x – h)) + k, shift the graph right by h and up by k
Apply vertical stretch or compression using a
Apply horizontal stretch or compression using b
Move the asymptotes to x = h + π/(2b) + kπ/b
Move the x-intercepts to x = h + kπ/b
Sketch each branch between asymptotes with the same increasing shape