Week 4 - Trigonometric Functions

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MOST of the things we're asked to graph, or the things that are useful to graph, are functions. Sometimes those functions are lines, sometimes they're parabolas, and sometimes they are very strange looking curves that take different paths depending on which part of the domain you're looking at (piecewise functions). Despite the huge number of different kinds of functions, some things are true about all functions; among them, that you can always plug in a number, get a number out, and then plot that number on a coordinate plane. Do that enough times, and you'll have the graph of a function. With that comes the idea of transformations. If you subtract 2 from a function, the effect is that you've subtracted 2 from the OUTPUT of a function, and consequently moved the entire graph down 2 units on the coordinate plane from where it would have normally been. We've talked about transformations before, and for the current unit on trigonometric functions, all of the same rules apply. The only caveat is that these are the first "periodic" or "circular" functions that we've seen, so there is some (very) slightly different terminology and some standard variables that we use to represent different things.

The concept of angular velocity, ω, and how it determines the period of a function is one of the most important things to grasp. Probably the most common transformation of sinusoidal functions (sine and cosine) in the real world are changes to the period. This shows up in everything from electrical signals to sound waves. When we compare this to our normal vocabulary for functions, the ω on a periodic function is the same thing as when we multiply the input of a function by a number, which has the effect of a horizontal stretch or a horizontal compression. That is, of course, not the only kind of transformation we can make to a function. We can also do vertical stretches (with periodic functions this effects something called the "amplitude"), and we can translate the function left or right and up or down. Horizontal translations of trigonometric functions (or any periodic function) is called a "phase shift".

The above equation is what can be considered the general form of a trigonometric function, with commonly used variables representing the different kinds of transformations that are possible. A is amplitude of the function, which is the distance from the mid-line to either a maximum or a minimum (alternatively, 2A is the "height" of the function from a minimum to a maximum). ω is the angular velocity of the function, which effects the period. The fundamental period of the sine function is 2π, and to determine the period (T) of a given sine function we simply divide 2π by ω, or T=2π/ω. φ (phi) is the greek letter we usually use to represent the phase shift (horizontal translation) of a trigonometric function. In the way it is shown above, φ is the actual distance left or right that a given point on the graph will translate. B is the vertical shift of the graph, and can be used to locate the mid-line of the function. Note that our independent variable when working with trigonometric functions is really an angle value, so in the function above we're using the greek letter theta (θ) to represent the input. It is very common to just use x (as seen in the video above) as the input for simplicity.

If we were given the equation of a trigonometric function, how can we generate the graph?

• Practice problems for graphing sine functions: PG 134 #13, 26, 33, 37, 39, 43, 53 (for the graphing ones, set them up as shown in the video (find the amplitude, period, phase shift, vertical shift, and lay out the tick marks)

It is possible to divide the period of each trigonometric function into 4 equal parts. After using ω to find the period of a function, we can then divide that by 4 to determine how far apart the 4 equal parts are. If the function has a phase shift, we can start plotting points along the horizontal axis, starting at the phase shift, to represent the 4 equal parts. If the function has a vertical shift, we can essentially just "shift" the x-axis up or down to that value. Then, by using our knowledge of the general behavior of the trig functions, we can incorporate the amplitude and plot out 5 important points on the graph and indicate that that patterns repeats indefinitely.

If we are given the graph of a function, how can we find the function?

• Practice problems for finding the equation given the graph: PG 134 #67, 70, 71, 72, 76

The information needed to write the equation of a trigonometric function is the amplitude, ω (from the period), the phase shift, and the vertical shift. We can systematically pull all of this information off the graph of a trig function. The Amplitude is the vertical distance between the mid-line of the function and its maximum. You can alternatively divide the difference between the maximum and minimum values and divide by 2. The vertical shift can also be found using the minimum and maximum values (or just from the mid-line of the function if that is apparent). The vertical shift can also be though of as the "average height" of the function. So, we can add the height of the maximum to the height of the minimum and divide that by 2. If you can determine the distance between consecutive minimums or consecutive maximums of the function, you've found the period. Since we know that T=2π/ω, we also know that ω=2π/T. We can then use our observed period, T, to calculate ω. The last thing you can pull from the graph is the phase shift. For a sine function, look at the start of the period closest to the y-axis and compare that to where the period would normally start (the y-axis if you're looking at the mid-line).

All of the above information applies to all trig functions (with some slight terminology changes), but really directly applies to cosine. Sine and cosine are extremely closely related, and in practical use, cosine is actually much nicer to use. We will dive into cosine and the other trig functions next week.

Hey, do this google forms activity so I can get a sense of how well you understand the material from this week! (Due by Monday 4/27 at 11:59PM).