For trig functions there are some important quantities that give us information about the shape of the graphs, and we should figure out how the various transformations effect these quatities.
- Wave Quantities
- Amplitude
Sine and cosine look like waves. Amplitude tells you how large those waves are. Like waves, the graphs have high points, the crests or peaks, and low points, the troughs. Amplitude is half total height difference between a crest and a trough. Alternatively, amplitude is the height of a crest compared to the average height of the wave, which is the same as the depth of a trough compared to the average height of a wave. For sine and cosine the amplitude is 1. Using the first definition, the crests have height 1 and the troughs have height -1, so the difference is 2 and half of 2 is the amplitude, 1. Alternatively, the average height is zero, and the crests are height 1 abouve zero while the troughs are height 1 below zero. - Period
The trig functions are periodic. Sine and cosine are
-periodic:
The same is true for secant and cosecant.
Tangent and cotangent are
-periodic:
The period of a trig function is the minimum a such that for all
we have 
.
If you have seen waves in other contexts, like a Physics class, you are also probably familiar with the notion of "frequency" of a wave. Period and frequency contain the same information about a wave.
They are inverses of each other. Frequency tells you number of cycles per second while period tells you number of seconds per cycle.
- Phase
Phase is a concept of horizontal shift for a wave. There is some disagreement about how to define phase shift. We will follow the definition in the text, which says that
has a phase shift of
.
You may have guessed that the phase shift was $c$, and sometimes it is defined that way, but it is important to notice that
does NOT look like the graph of 
shifted horizontally by $c$.
To shift 
horizontally by $c$ we would replace $x$ with 
, giving:
Here are some examles:
The black curve is the graph of
.
The red curve is the graph of
.
The green curve is the graph of
.
The red curve is obtained by shifting the black curve to the right by
. It makes sense to say that the graph of the red function has a phase shift of 
.
The red curve is the graph of 
, it does not have an obvious 
in it, we ge the 
by taking 
.
The green curve is obtained by shifting the black curve to the right by
. The green curve is the graph of 
and we get the 
by taking the 
in the formula and dividing by 2.
- Amplitude
- Standard Transformations
- Vertical Transformations
Applying operations outside of the trig function produces a vertical change. Adding a constant gives a vertical shift, multiplying by a constant gives a vertical stretch. For example the graph of
would look like the graph of 
, but would be stretched vertically by a factor of 2 and then the whole graph would be shifted up 4.
These transformations only change the graph vertically, so phase and period are unaffected.
Amplitude is affected by the multiplication, but not by translation.
has amplitude 1, so 
has amplitude 2, the crests are twice as high and the troughs are twice as low.
also has amplitude 2. Now the crests go up to 6, but the troughs only go down to 2 so the difference is 4, and half of that difference is the amplitude.
Important: You can not determine amplitude just by looking at the maximum height in a graph, you have to consider maximum change. The amplitude of
is 2, even though the maximum height of the graph is 6!
- Horizontal Transformations
Multiplication inside the function produces horizontal stretching. Consider
. Multiplication by 
causes the 
curve to oscillate faster (for 
).
This changes the period.
I recommend that you DO NOT try to memorize how the period changes. Figure it out.
has a period of 
. This means that 
will complete one cycle as its input varies from 0 to 2π. The input is now 
.
when 
. 
when 
. So 
completes one cycle as 
goes from 0 to 
.
Thus the period of 
is 
.
We can achieve a horizontal shift by replacing
with 
. This requires some care if there is also a multiplication.
Consider 
again. To shift right by 
replace 
by 
and get 
.
It is important to understand that
is NOT 
shifted by 
. As we just said, to shift by 
you would need 
.
is only a shift by 
. See the discussion of Phase above for more examples of phase shift.
- Vertical Transformations
and the finish line is at 
. (It would be hard to tell from the picture exactly where these lines should go, but suppose we know that our curve crosses the equilibrium line at exactly 
and 
.) The distance between these two lines is the period, so in this case the period is
.
and 
.
From the period we have:
and since 
that means 
.
, so we draw horizontal lines at heights 
and at 
.
These are the max and min lines.
coordinate equal to the phase shift, in this case at 
.
-coordinate equal to the phase shift plus the period, so at 
.
