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Graphing Trig Functions: Examples (page 2 of 3)

Sections: Introduction, Examples with amplitude and vertical shift, Example with phase shift

  • Graph one period of s(x) = –cos(3x)
  • The "minus" sign tells me that the graph is upside down. Since the multiplier out front is an "understood" –1, the amplitude is unchanged. The argument (the 3x inside the cosine) is growing three times as fast (because of the 3), so the period is one-third as long; the period for this graph will be (2/3)π.

    Here is the regular graph of cosine:

      graph of cos(t) from zero to 2pi

    I need to flip this upside down, so I'll swap the +1 and –1 points on the graph:

      graph showing the new points relative to the 'regular' graph

    ...and then I'll fill in the rest of the graph: Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

      graph of -cos(t) in blue, with original graph in gray

    And now I need to change the period. Rather than trying to figure out the points for the graph on the regular axis, I'll instead re-number the axis, which is a lot easier. The regular period is from 0 to , but this graph's period goes from 0 to (2π)/3. Then the midpoint of the period is going to be (1/2)(2π)/3 = π/3, and the zeroes will be midway between the peaks and troughs. So I'll erase the x-axis values from the regular graph, and re-number the axis:

      graph of -cos(3t), showing same line but with horizontal axis re-numbered

Notice how I changed the axis instead of the graph. You'll quickly get pretty good at drawing a regular sine or cosine, but the shifted and transformed graphs can prove difficult. Instead of trying to figure out all of the changes to the graph, just tweak the axis system.

  • Graph at least one period of f(θ) = tan(θ) – 1
  • The regular tangent looks like this:

      graph of tan(theta) from 0 to 2pi

    The graph for tan(θ) – 1 is the same shape, but shifted down by one unit. Rather than try to figure out the points for moving the tangent curve one unit lower, I'll just erase the original horizontal axis and re-draw the axis one unit higher:

      same graph, but with the horizontal axis drawn one unit higher, thus shifting the graph down one unit

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Cite this article as:

Stapel, Elizabeth. "Graphing Trig Functions: Examples." Purplemath. Available from Accessed


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