Transformations of Trig Graphs
Before reading through these notes and filling them in, read in your book in section 6.5 pages 547 through 554 about
how to get the basic shapes of the graphs of the 6 trig functions. Make sure you fill in everything in these notes. If you
don’t understand something look at the solution and fill in the notes anyway, but make sure and ask about it at our next
class.
Go to this website:
http://laurashears.info/math122/unit2/graphingTransformedTrigFunctions/ and
fill in the following notes as you work through the pages of the online notes.
First review the shapes of the 6 basic trig functions by drawing them.
General Directions:
Let y = a trig(bx + c) + d represent a trigonometry function where 'trig' stands for any of the six functions. First
find the period, phase shift, vertical shift, and vertical stretch (called the amplitude in the case of sine and
cosine) of your trig function.
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period = period of the original trig function / b
phase shift = -c/b (Notice that the phase shift is simply the value of x that makes the argument equal to
zero.
vertical shift = d
vertical stretch (amplitude) = a
Note: There is also a vertical reflection through the line y = d if a is negative.
Example 1: What is the period, phase shift, amplitude, and vertical shift of y
= 3sin (2x + π) ‐ 1? Also, draw a rough
sketch of the reference graph of y = sinx to keep in mind while drawing the transformed graph.
1.
2.
3.
4.
5.
6.
7.
Since our period and phase shift are both in terms of π, let's choose a horizontal scale in terms of π and an integer scale for
the vertical scale. Mark the scale on the grid in your notes or print the one below and mark it on it.
Next use the phase shift and the vertical shift to find where the point that was at the origin moved to and plot it on the
graph.
Next move horizontally a period away and plot another point. Repeat this until you run out of room.
Next add the points that are halfway between the endpoints of a period.
Next add the quarter of the way through a period points.
Next add the three-quarter of the way through a period points.
Finally, draw a nice smooth curve through the points in the shape of the trig graph.
Example 2
y = 2cos(πx/3 + π/2) + 1.
First identify the four characteristics: amplitude, vertical shift, period, and phase shift. Also draw a reference
graph for cosine. Then click 'Next'.
1.
2.
3.
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6.
Next pick a good scale to go with the problem. Use the period and phase shift to determine your scale. Since neither of
them are in terms of π, your scale should not be in terms of π either.
Next, plot the point that corresponds with the start point of the cosine graph, (0, 1), taking into consideration the phase
shift, verical shift, and the amplitude since cosine starts at a high point.
Next, plot all of the rest of the high points. Note that they are multiples of a period away from the one that we already
plotted.
Halfway between all of the high points are low points, so plot all of the low points.
Halfway between the high points and low points are the middle points. Plot those.
Draw a nice smooth graph through your points in the shape of a cosine curve.
General Plotting guidelines:
1. Use the period and phase shift to decide what kind of scale you need on your axis (i.e. a scale in terms
of rational numbers or in terms of π).
2. Label at least one tick mark on each axis. Then each tick mark stands for the same distance.
3. Keep in mind what the basic shape of your trig function looks like. It is helpful to draw a reference
graph.
4. Let x = phase shift and y = a trig(0) + d. Plot this point first.*
5. Move one period to the right (and/or left) and plot a point at the same y value.* Repeat this process
for as many periods as you can fit on your grid.
6. (In the horizontal direction) go halfway between the two endpoints of each period that you are
plotting. Plot that x value with its corresponding y value* (y = atrig(per/2)+ d).
7. (Also in the horizontal direction) go one-fourth and three-fourths of the way between your endpoints
(i.e. halfway between the points you already have plotted). Plot these points with their
corresponding y values* (y = a trig(per/4) + d and y = atrig(3per/4) + d respectively).
8. Now draw the trig function.
*When you have an asymptote instead of a point at an y value, you will draw a dotted vertical line instead of
plotting a point.
Example 3
f(x) = 3tan(x/4 − π/2)
Find the characteristics: vertical stretch, vertical shift, period, and phase
shift of f and draw the reference graph for y = tan x.
1.
2.
3.
4.
5.
6.
Pick good scales for your vertical and horizontal axis.
Find your start point. In other words, where does the point that is originally at x = 0 go after the transformation?
Plot your other points that are integer multiple of periods away from your start point.
Go half way between your period start points and plot what belongs there.
Next plot the points that are halfway between your start points and your vertical asymptotes.
Finally, draw your graph.
After trying a few standard graphing problems, you can read about the special graphing cases on pages 567 (starting at
the bottom where it says, “Graphs of Sums: Addition of Ordinates” through the bottom of page 569. You will find some
clarification of these concepts at: http://laurashears.info/math122/unit2/spec_grph_cases/