How To Graph Trig Functions

Format:

|a| = amplitude, b affects period, b & c affect phase shift, and d is the vertical shift

1.  Label the x-axis.

  1. Calculate the period: ______

For sin x, cosx, cscx, secx, period = ; For tanx and cotx period =

The period is the length of one cycle.

  1. Calculate = ______You will label 4 critical points through the period, so divide the period by 4.
  2. Draw 8 marks on the x axis. Label them 1, 2, 3, 4, 5, 6, 7, and 8 times your answer in part b and reduce. For example, if =, you would write: . These would reduce to

2.  Label the y-axis.

  1. If you don’t have a vertical shift (no d), just label a and –a on the y-axis.
  2. If you have a vertical shift, draw a dashed line though d. Then add and subtract a from d and label those numbers on the y axis as well.

3.  For sine and cosine graphs, sine starts at 0. Cosine starts at a. The pattern is 0, max, 0, min, 0, max, 0, min, 0…. If a is negative for sine, still start at 0, but go down first. Remember that a negative coefficient flips the graph.

4.  For secant and cosecant graphs,

  1. Graph the reciprocal graph with a dashed line. For secant, graph cosine, and for cosecant, graph sine.
  2. Where the zeros are in your pattern, draw vertical asymptotes. You can’t take the reciprocal of zero!
  3. Draw U-shapes between the asymptotes with the vertex at the max and min values. The asymptotes define the width, so draw your U’s close to the asymptotes, but don’t cross them!

5.  For tangent and cotangent graphs,

  1. KNOW YOUR PARENT GRAPHS!!!! You can graph sine and cosine because you know your patterns. All tangents and cotangents look the same, so learn the patterns! For tangent, it is 0, a, asymptote, -a, 0, a, asymptote. For cotangent, it is asymptote, a, 0, -a, asymptote. Change your x and y axis values using the directions above. Tangent is strictly increasing. Cotangent is strictly decreasing.
  2. Tan(0) = 0 and cot(0) = undefined or asymptote, so tangent starts at zero & cot(x) starts at an asymptote.
  3. Tangent and cotangent are reciprocals of each other. The reciprocal of zero is undefined. The reciprocal of the undefined values are zero.