How to Graph Trig Functions?

Identify the function type: sine, cosine, tangent, cotangent, secant, or cosecant

Write the function in transformed form: y = a trig(b(x – h)) + k

Find the amplitude: |a| for sine and cosine

Find the period:

Sine and cosine: 2π / |b|

Tangent and cotangent: π / |b|

Secant and cosecant: same as the related sine, cosine, tangent, or cotangent function

Find the phase shift: h

Find the vertical shift: k

Find the midline: y = k

Find reflection:

If a < 0, reflect over the x-axis

If b < 0, reflect horizontally

Plot the key points for one period

Use the parent function’s pattern:

Sine: starts on the midline

Cosine: starts at a maximum or minimum

Tangent: crosses the midline and has vertical asymptotes

Apply the period to space the key points evenly

Apply the phase shift to move the graph left or right

Apply the vertical shift to move the graph up or down

Draw smooth curves through the points for sine and cosine

Draw asymptotes and branches for tangent and cotangent

Draw branches approaching asymptotes for secant and cosecant

Label important points, asymptotes, and intercepts

Check the graph against the amplitude, period, shifts, and reflections