You've Already Learned the Basic Trig Graphs

Transforming Trig Graphs

We can transform and translate trig functions, just like you transformed and translated other functions in algebra.

Let's start with the basic sine function, f(t) = sin(t). This function has an amplitude of 1 because the graph goes one unit up and one unit down from the midline of the graph. This function has a period of 2π because the sine wave repeats every 2π units. The graph looks like this:

Now let's look at g(t) = 3sin(t):

Do you see that the graph is three times as tall? The amplitude has changed from 1to 3. This is always true: Whatever number A is multiplied on the trig function gives you the amplitude; in this case, that number was 3. So 0.5cos(t) would have an amplitude of 1/2, and –2cos(t) would have an amplitude of 2and would also be flipped upside down.

Do you see that the graph is squished in from the sides? Do you see that the sine wave is cycling twice as fast, so its period is only half as long? This is always true: Whatever value B is multiplied on the variable, you use this value to find the period ω (omega) of the trig function, according to the formula:

In the sine above, that value was 2. (Sometimes the value of Binside the function will be negative, which is why there are absolute-value bars on the denominator.) The formula says that cos(3t) will have a period of (2π)/3 = (2/3)π, and tan(t/2) will have a period of (2π)/(1/2) = 4π.

(Note: Different books use different letters to stand for the period. In your class, use whatever your book or instructor uses.)

Now let's looks at j(t) = sin(t – π/3):

Do you see that the graph (in blue) is shifted over to the right by π/3 units from the regular graph (in gray)? This is always true: If the argument of the function (the thing you're plugging in to the function) is of the form (variable) – (number), then the graph is shifted to the right by that (number) of units; if the argument is of the form (variable) + (number), then the graph is shifted to the left by that (number) of units. So cos(t + π/4) would be shifted to the left by π/4 units, and tan(t – 2π/3) would be shifted to the right by (2/3)π units. This right-or-left shifting is called "phase shift".

Now let's look at k(t) = sin(t) + 3:

Do you see how the graph was shifted up by threeunits? This is always true: If a number D is added outside the function, then the graph is shifted up by that number of units; if Dis subtracted, then the graph is shifted down by that number of units. So cos(t) – 2 is the regular cosine wave, but shifted downward two units; and tan(t) + 0.6 is the regular tangent curve, but shifted upward by 6/10 of a unit.

Putting it all together, we have the general sine function, F(t) = Asin(Bt – C) + D, where Ais the amplitude, Bgives you the period, Dgives you the vertical shift (up or down), and Cis used to find the phase shift. Why don't you always just use C? Because sometimes more is going on inside the function. Remember that the phase shift comes from what is added or subtracted directly to the variable. So if you have something like sin(2t – π), the phase shift is not πunits! Instead, you first have to isolate what's happening to the variable by factoring: sin(2(t – π/2)). Now you can see that the phase shift will be π/2 units, not πunits. So the phase shift, as a formula, is found by dividing Cby B.

For F(t) = Af(Bt – C) + D, where f(t) is one of the basic trig functions, we have:

o A: amplitude is A

o B: period is (2π)/|B|

o C: phase shift is C/B

o D: vertical shift is D

Let's see what this looks like, in practice, because there's a way to make these graphs a whole lot easier than what they show in the book....

· 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:

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

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 2π, 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:

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:

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:

· Graph two periods of g(x) = sin(πx + π/2) + 3

The amplitide is going to be the same, but the midline of the graph is going to be at y = 3 instead of y = 0 (that is, the x-axis). The period is going to be (2π)/(π) = 2. The argument factors as π(x + 1/2), so the graph will be shifted 1/2 unit to the left.

Instead of shifting the sine wave three units up, I'll shift the horizontal axis three units down, and re-number the y-axis:

The regular period goes from 0 to 2π; this one goes from 0 to 2, so I'll re-number the x-axis:

And the graph is shifted to the left by 1/2, so I'll shift the y-axis to the right by 1/2 and re-number the x-axis a bit more:

Can you see why I used pencil and did a lot of erasing when I was doing graphing?

My best advice regarding doing these graphs is to practice, practice, practice. You don't want to freeze or have a "brain-fart" on the test, and this is pretty straightforward, once you get the feel of it. So keep doing extra graphing, until you are feeling comfortable and confident in your skills.