Identify the basic form: (y=atan(b(x-h))+k)
Find the period: (pi/|b|)
Find the phase shift: (h)
Find the vertical shift: (k)
Find the vertical stretch or compression: (|a|)
Determine whether the graph is reflected over the x-axis: if (a<0)
Draw the midline: (y=k)
Locate the center point: ((h,k))
Find the vertical asymptotes: (x=hpm frac{pi}{2|b|})
Mark additional asymptotes every (frac{pi}{|b|}) units
Plot key points between asymptotes:
(x=h-frac{pi}{4|b|})
(x=h)
(x=h+frac{pi}{4|b|})
Use the points:
(left(h-frac{pi}{4|b|},,k-aright))
((h,k))
(left(h+frac{pi}{4|b|},,k+aright))
If (a<0), reverse the direction of the curve
Sketch the tangent curve between each pair of asymptotes
Repeat the pattern every period
Check that the graph approaches each asymptote
Label the asymptotes, midline, and key points
Use radians for accurate graphing
For (y=tan x), use:
Midline: (y=0)
Asymptotes: (x=pm frac{pi}{2}+kpi)
Key points: (left(-frac{pi}{4},-1right)), ((0,0)), (left(frac{pi}{4},1right))