Updates on the Status of the Graphing iPad Quiz

Hey Trig! I'm sorry for all of the confusion surrounding the iPad quiz. For 2nd and 4th periods: we will be ready to go for Monday so bring your 1/2 page cheat sheets and be ready to record. 6th Period: you were fantastic with those quizzes and I was genuinely impressed by the quality of work that you submitted. Well done! :)

Practice for Graphing Quiz, Day 1

We practiced graphing three trig functions on the iPads today. Your goal on the quiz is to be able to graph a function while explaining what you are doing while you work, and a brief explanation on why you're doing it. If you want additional practice, try going to the online website that we used today:

awwapp.com

which is an online whiteboard. Here, you can graph trig functions (or anything else, for that matter) and practice without killing the rainforest. Hooray!

Videos of me graphing and explaining trig functions

http://www.educreations.com/lesson/view/graphing-trig-function-example/12573134/?s=0Zzdf5&ref=app

http://www.educreations.com/lesson/view/graphing-horizontal-translations/12573232/?s=XYVdvD&ref=app

Two GREAT examples of how to graph trig functions with multiple numbers in them! Note: you must have the "Educreations" app downloaded to view these tutorials. to do this, just go to your App Store and search "Educations" and you can download it for free. You'll have to create a free account and then you can view all of my free tutorials!

Horizontal Translations of Trig Functions

This is our last type of change to a base trig function! We see horizontal translations when there are parentheses in the function, such as in f(x) = sin(x + 2/3pi). You will usually see the term inside the parentheses in radians, and since we don't like radians, you'd want to change it to degrees first then re-write the function using degrees: f(x) = sin(x + 120).

Horizontal translations change the start and end point of the function on the x-axis. We know that sine/cosine normally start at 0 and end at 360. So we form an inequality: 0 <= x + 120 <= 360 and solve to get x by itself in the middle by subtracting 120 from everything. This gives us -120 <= x <= 240. So, now the function begins at -120 and ends at 240! From there, find your new midpoints and graph your function.

Note that horizontal translations ONLY change the start and end points. They do not change the amplitude, period, direction, or starting point on the y-axis. There may be other numbers in the function that do that, but anything inside the parentheses is a horizontal translation only.

More Challenging Vertical Translations of Trig Functions

We see a vertical translation when there is a number behind the x-value in the function. However, there may be multiple numbers in the function that we need to address.

For example:
y = -2sin4x - 3.

All of the numbers in the function do different things to change sinx from its base function position.

-2, since it is in front of the sin, changes the amplitude of the function. Normally sine has an amplitude of 1, but this makes the amplitude -2. Instead of going up first, the negative flips it, and makes it go down first.

4 changes the period since it is in front of the x. We start with 2pi (always), and divide it by 4 to get the new period. Since we would get 1/2pi, convert this from radians to degrees by multiplying by (180/pi). This will give you a new period of 90 degrees.

-3 is the vertical translation since it is at the end of the function, behind the x. This means that instead of starting and ending at 0, as sin usually does, the function starts and ends at -3 on the y-axis.

Vertical Translations of Trig Functions

A vertical translation means that we move a base function (a function with no numbers) up or down on the y-axis. In other words, we change the starting and ending points for the y. The amplitude, period, and degrees on the x-axis stay the same - only the y-axis changes.

We see a vertical translation whenever there is a number added or subtracted at the end of the function. For example: y = sinx + 2 means that we would take the basic graph of y = sinx and move the whole function up 2 units on the y-axis. Instead of starting at (0, 0), we would start at (0, 2).

For cosine, it's a bit trickier because the graph of cosine begins at (0,1) instead of (0,0). Therefore, for the graph of y = cosx + 2, we would start at positive 3 instead of positive 2 because cosine already starts at 1, so we have to move the graph up two units from where it already starts.

IMPORTANT NOTE: just because we have a vertical translation, does not mean the amplitude of the function changes. Unless we see a number in front of the function, the amplitude would still be 1. However, if we do see a number in front of the function, such as y = 2sinx + 2, then we would have a vertical translation as well as a change in amplitude (because there is a number in front of the trig function).

Period of Sine and Cosine

We know that the base function of sine and cosine goes from 0 - 360 degrees. When we change the period of the function, we are changing how long/short the function is before it repeats itself again. So instead of going from 0 - 360, they would be going from 0 - some other number.

We know that a change in amplitude can be found be y = 2sinx where the 2 in front of the sin shows how high the new function goes. For a change in period, the number is instead directly in front of the x, such as y = sin2x. To get the new period, we know that 360 = 2pi. Divide 2pi by whatever is next to the x: in my example, it would be 2pi/2. The pi is left over, so pi would be our new period. We would much rather work in degrees than radians, so convert pi to degrees by multiplying by 180/pi. The pi's cancel, and you have 180 degrees left. Thus, 180 degrees is the new period of pi. The function still starts at zero, still ends at zero, and still has the same shape, but it becomes only half as long as before. The same rules apply for cosine functions, too!

Note that the amplitude doesn't change if there is no number in front of the trig function. Our function still goes from [-1, 1], but now, it just repeats itself more often because it's period is shorter.

Thursday and Friday: Amplitude for Sine and Cosine Functions

Now that we know what the base function for sine and cosine looks like, we can manipulate/change that base function in many ways. The first way is to change the range of the function by stretching/compressing it: this is called a change in amplitude.

When changing the amplitude of the function, we would expect to see a number in front of the trig function, like this:

y = 4sinx
y = (1/4)sinx
y = -4sinx

and the same applies for cosine functions as well. The "4" in the first example stretches the normal range of sine [-1, 1] to [-4, 4]. The "1/4" in the second example compresses the normal range of sine [-1, 1] to [-1/4, 1/4]. And the "-4" in the third example both stretches the function, like in the first example, and flips the function upside-down.

We practiced graphing on Thursday and made artwork out of five trig functions on Friday, where Mr. Williams will be judging for the top three on Monday. Shout out to everyone who brought their creative side to class on Friday! (and special shout out to Quadarius Gatewood, who found the creative side that he didn't think he had in a matter of 15 minutes!)

Base Functions for Sine and Cosine

Not only can we use sine and cosine to find missing sides of a right triangle, or points on the unit circle, but we can also graph them as functions of x and y on the coordinate plane. We graph them using degrees found on the unit circle. Sine's base function is y = sinx, and cosine's base function is y = cosx, and they look like this:

Both sine and cosine have infinite domains from negative infinity to positive infinity. Both of their ranges are from -1 to 1. And both repeat themselves after 360 degrees. When a function repeats, we call it a periodic function. Both of the periods of sine and cosine are from 0 - 360. The only difference between sine and cosine are that sine begins and ends at 0, while cosine begins and ends at 1.

Guest Speaker and Meeting out of Nowhere

We had a guest speaker yesterday from Arkansas State University! Shy the master's student told us about the Oral Communications class that she teaches to college freshman, and what it's like to be a freshman on the A-State campus. Expectations are a lot different in college - especially when your class only meets one day a week but you have it for 3-hours at a time!

I was pulled from my second period class today (just as we were about to graph y = sinx - boo!) to attend a training session at the Junior High. My apologies for this one - I didn't know about it in advance. A BIG THANK YOU to everyone who practiced their math ACT while I was gone and a HUGE Shout Out to my aid Bre Womack for setting everything up while the sub was there!

Error Analysis

All of my classes are close to being caught up - thank goodness! While 6th period was catching up on Friday, 2nd and 4th period did an exact trig values error analysis (shout out to 4th period for doing a great job explaining it when Mrs. Long walked in!). We looked at problems and found where the mistake was being made in each. Then you said how the problem should be fixed, and then fixed it to find the correct solution.

I have most of the review sheets. If I don't have yours yet, please get it to me so that you can at least get partial credit (remember, report cards come out soon, don't make a zero right before report cards!).

Negative Exact Values of Trig Functions on the Unit Circle

Finding a negative exact value of a trig function is very similar to solving when an angle is greater than 360 degrees.

We know that the unit circle begins at zero degrees and goes all the way up to 360 degrees. For anything outside of that range, we must use properties of coterminal angles to solve and find the corresponding degrees/coordinates on the unit circle.

Suppose that you were given an angle of -120 degrees. Since that's not on the Unit Circle, we would add 360 and finding its corresponding degrees and coordinates. We could then use those (x, y) coordinates to find the exact values of sine, cosine, tangent, cotangent, secant, and cosecant at that degree.

Review sheets due tomorrow for everyone except 6th period - yours are due Monday.

Cosecant, Secant, and Cotangent Exact Values, and Coterminal Exact Values

Yesterday, we discussed how to find exact values of sine, cosine, and tangent using values on the unit circle. Today, we did the same thing except with their inverses: cotangent, cosecant, and secant.

Since we know that, in a coordinate pair (x,y)
 sine = the y value
cosine = the x value
tangent = y/x

We can use the inverses and say that:
cosecant = 1/y
secant = 1/x
cotangent = x/y

Sometimes, the radian that we convert to won't be located within the 0-360 degrees that we see on a unit circle. For example, 13pi/3 converts to 780 degrees, so that would be way off of 360, which is the last degree measure on the unit circle. In this case, we must use properties of coterminal angles to solve. Simply subtract 360 until you get a measurement that is on the Unit Circle (in this case, 780's coterminal angle would be 60 degrees). Then use the (x, y) coordinates for the coterminal angle instead.