Precal Blog

So 6th hour had Mr. Anderson again aannd we started off the class by switching our homework with somebody else and they tried to figure out equations from the graphs that we drew for them. They also had to find the amplitude, equation of a midline, period, 1/4 of a wavelength, and the five critical points for one full period.

So I guess I will start off with a little simplified review of what Britney posted yesterday and what the homework was all about…

• mideline cuts your graph in half (horizontally…)
• amplitude is how far away the high/low point is from the midline.
• period is how long it takes you to complete the graph before it repeats itself, for example a clock takes 12 hours before it begins another round. In an equation…for example f(x)=2sin(6x)+1 , the amplitude would be the number right before the sin. If you are multiplying this by 6 like in the equation above, you will be making the distance that it takes to be a full rotation 1/6 its size (making the peaks on the graph 6 times closer together).
• a fourth of a wave, simply divide 360 by 6, and divide that number by 4. This will be 1/4 of a peak, which you will need to find critical points on your graph.

GRAPHING COSINE

We learned a lot about using cosines in the periodic functions which was not too difficult because its almost exactly like using sine but with a few minor differences. One of these is that when you take the cosine of 0, you get 1.

When looking at a graph, this means that the cosine graph will be exactly one unit away from the sine graph.

Mr. Anderson also told us that if you wanted to look at a certain part of the graph, you just restrict the domain… for instance if only a certain part of a graph made sense for the problem. Kind of like prices for cars only make sense when you look at the time they were priced.

We then looked at how an equation such as f(x)=cos(x+30), (x representing theta), would change the f(x)=cosx graph.

In all equations like this, adding a number inside the parenthesesmoves the graph left or right…in this case it would move it 30 to the left. This means that the high points on the graph will be shifted 30 to the left. (0,1) is the relative maximum on the regular cosine graph, on the new one, the maximum point is (-30,1).

The new graph will finish 30 degrees after 360.

Mr. A then showed us that Sin & Cos are simply translations of each other. To make the cos. graph the same as the sin., use the equation cos(x+90). Any multiples of 360 from 90 will make the graphs equal, such as cos(x+270).

The final thing he showed us for the day was that you can sometimes factor out numbers inside the parentheses of the equation. For example…f(x)=sin(2x+90) it might be easier to look at this equation as …f(x)=sin(2(x+45). Because you are doubling the angle, the period will finish in half the time.

He then gave us the homework for tomorrow which is …

Exploration 3-2a: Transformed Sinusoid Graphs

THE SCRIBE FOR TOMORROW, NOV 1st WILL BE……..

Rose

Because Mr. B is still out for his shoulder, Mr. Anderson ran our class today. Sitting down in our freshman math room, was like a trip down memory lane, and it was very helpful having a math teacher, instead of another substitute, to keep us updated on our lessons.

To start off class, we turned in our claimable problems from Sections 2.4 and 2.5.

Next, Mr. Anderson reviewed some basics about translations to get a feel for what we as a class already knew. We quickly reviewed what it meant to place values inside the parenthesis, outside the parenthesis, and also what it meant to add values to the function.

This was all a review for exploring how these same translations can be applied to the graphs of sin and cos, and how they can be interpreted.

The examples we did during class today involved sin graphs.

We reviewed that the domain of a sin graph is any number, and the range of a sin graph is between -1 and 1.

We started off by looking at the basic function of sin, f(x)=sin(x) and talked about different characteristics about the graph.

We learned that a sin graph is an example of a Periodic Function. A Periodic Function is a function that repeats itself, such as a clock, working as a cycle over and over again. In the case of a sin graph, f(x)=sin(x) has a Period of 360 degrees, because every 360 degrees, the graph repeats itself.

We then were told that there are Five Critical Points to every sin graph.

The Five Critical Points f(x)=sin(x)

THE FIVE CRITICAL POINTS ARE SHOWN IN THE LINK ABOVE.

The five points exist on ONE period, which is also show below. These points divide the period into four equal sections.

The Inflection Point seperates the concave down part of the graph from the concave up. This is the point where the graph literally flips, or reflects the first part over the x axis.

For the graph above, y=0 is the sinusoidal axis, or midline. This line “breaks” the y values of the graph in half and is located directly in the middle, where the graph begins to reflect. The values above and below this line to the high and low points of the graph, are equal, and are referred to as the Amplitude of the graph.

For example, the amplitude of the graph f(x)=sin(x) would be 1 because the y=o is the midline, and the high and low points of the graph are each exactly 1 unit below this line.

These new terms can also be seen and explained on Page 87 of our book.

Next we did some transformations of the sin graph, and analysed how these transformations effected the midline, the amplitude, and the period.

After a few examples, and manipulating of our functions, we were finally able to understand a few things about sin graph transformations, and how really, they were not all too different from transformations we had earlier performed.

For example, the equation: f(x)= -2sin(4x)+3

• would cause the first “hump” of the sin graph so curve downward (because of the -2)
• would cause the amplitude, or slope distance of the graph to be 2
• would have a midline of 3
• and would have a Period of 90, because 90 is one FOUTH of 360 and so the period would be completed 4 times as fast as a normal, basic sin graph

We were then given a worksheet on sin graphs and their functions, doing the same exact thing–interpreting the equation from the graph and vice versa.

We were also told that Mr. B wanted to remind us to work on our gold sheet for 3.2

and that the end of the quarter test and journal due dates are coming near.

The Test will be November 7th. And the journals will be due on November 8th.

So it is really a good idea to look over those requirements for the quarter journal.

TOMORROWS SCRIBE WILL BE….. MEGHAN E.

Thank you,

Britney.

Today in math we had Mr. Steeno as our subsitute. So we actually had to learn something in class today instead of like yesterday having a study hall for the hour.

To begin class today Mr. Steeno started off by writing on the board our homework for tonight . Which by the way for those who weren’t paying attention its a yellow sheet for section 3-2 and writing 4 equations for the packet he gave us at the end of class. Also the warning that the dreaded end of the quarter test is going to be on November 7 and the following day the quarter journal will be due November 8 for those who have forgotten about that.

Continuing on today in class before we started taking notes Mr. Steeno collected the presentation problems we were told to write out for the claimable problems. After the presentations were collected we began todays lesson.

The lesson pretty much discussed how to graph the trig functions…well at least sin and cos. We also discussed how the graphs would look and what you can figure out about the graph with or without the equation of it. The first graph we were given was produced by the equation `y=sin theta`.

agraph axes( ); xmin=-6.5; xmax=6.5; xscl=(pi/4); plot(sin(x)); endgraph

I believe that should work for the graph if not if someone could create a graph to go along with this problem that would be great. Thanks. Anyways from the graph and the equation you can figure out the the graph passes through `(0,0)`. This is a periodic function because the graph happens at regular intervals. From there you figure out the parent function has a period of `360 degrees`. Other helpful things Mr. Steeno taught us was:

• the midline cuts the graph in half
• the distance above and below the midline is always positive, they can’t be negative
• the graph doesn’t go past 1 or -1
• the amplitude of the graph is 1
• the amplitude of a graph means the graph will go up to 1 and down to -1 but never past that.
• in general the amplitude of a graph is the distance from the x-axis to the highest or lowest point in the graph

Next we learned about the 5 Critical Points. The points are distinctly split up by the:

• starting point
• relative max
• relative min
• end of graph
• inflection point or midpoint

You can find the start point at the start of the graph on the y-axis, the relative max. point at the highest point in the graph, the relative min. point at the lowest point in the graph, the end of graph point at the end of the graph on the x-axis, and the inflection point in the middle of the graph. The end of graph point can actually be determined bye the period of the graph because that is where the graph ends. The inflection point can be found on the midline of the graph which helps making that point easier to find. But when it comes to determining where the 5 critical points are you use the `1//4` ~ or quarter wave to find out after how many degrees the next point will be. So going back to the example graph Mr. Steeno used to figure out the `1//4` you set up the equation `1/4=360/4` =90 degrees. That means that every 90 degrees you have one of the 5 critical points on your graph.

Over the class period Mr. Steeno showed us a couple more examples to make sure we understood what was going on. He went on to explain that the inverse of a graph will just flip over the x-axis, and depending on the shape of the graph you’re given you can figure out the equation, midline, amplitude, period, `1//4`, and where the graph starts, but mainly the equation. Well that pretty much sums up the lesson we learned today in class.

The scribe for Wednesday will be Amy.

So Mr. B’s absence left us with little to do again. Most students worked on the problems for 2-4, 2-5. The claims and presentations will be handed in Tuesday. I’m excited to have Mr. Anderson as a teacher again. He is a pretty cool dude. Last night. Brett Favre was way cooler though. Football aside, nothing much happened involving mathematics yesterday. Get ready to hand in those clear and concise problems on you loose leaf!

The next scribe will be… Britney C

Today, sadly, Mr. Bieniek was not here to teach us. So we blew the time working on our problems from sections 2-4 and 2-5, especially our claims. There was a lengthy note on the board, but we were told by our sub to ignore it. At the beginning of class, Mr. Steeno came over to make sure we would be alright. And so, assisted by all this guidance, we were all able to take a sort of study hall and trudge through our sections. Blessed be the half-day.

Tomorrow’s scribe will be…. Em-eye-lee.

Wednesday was a half day, so we only had 23 minutes to devoted to mathematics, and 3 minutes to get to class. Once everyone arrived our substitute teacher read us our instructions, which were to work with our 4 o’clock partners on a worksheet, then to work on our 2-4/2-5 problems and claims.

Monday’s Scribe will be Adam

Please make sure and look at the requirements for the Quarter Journal. The end of the quarter is near.

Also, if you want to “present” any of the claim problems from sections 2-4 or 2-5, Messers Anderson and Steeno will collect those first thing on Tuesday. Don’t hand anything in that would be a poor presentation. I can’t ask clarifying questions looking at a piece of loose leaf – you will have to be extra careful to be clear in your explanations.

Today in class, we had a sub because of Mr. B’s shoulder. All we did in class was a worksheet and watch the runners walking for state thing. Other then that, nothing very important happen.

Monday’s scribe is Grant,

Running place to place…

Mr. B started off class with a good note by reading a “anonymous” love letter to an unknown person. After the class caught their breath, Mr. B announced some unfortunate news. He announced that he won’t be gracing us with his presence because of shoulder surgery. He can hardly throw a ball, and that is quite the predicament considering he has a four year old son. So on the sole purpose of being a good father figure, Mr. B is having his shoulder fixed.

After the news, he gave us a worksheet for the rest of the class period. We got together with our twelve o’clock partner and took on the task of doing the worksheet. Once that was completed, we worked more on the book problems.

He then gave us the claims and explained how we have to do them while he is away recovering. For the problems you want to claim you have to submit the problems on loose leaf, in the form as though you were presenting it. Each problem you claim will count as both a claim and a presentation grade. They have to be finished by Tuesday were you will give the papers the substitute.

The claims for 2-4 are 13, 33e, 34d and 41.

The claims for 2-5 are 13 and 20.

1/2 Day Tomorrow!!!!!!! (Just another reason to Party!!!!!!!!!!!!!)

The next scribe is Luke

We started off the class period with him telling us that Mr. B would be absent because he needs to have surgery on his shoulder becuase of an event that had happened a few years back. He needs to fix his shoulder otherwise it hurts him to play sports or even throw a ball, which he wants to do with his son. He said he will be gone for a short while but if you have any questions ask one of the other math teachers, or leave a question on the blog for him.

After he had explained why he was going to be gone, he gave us a worksheet and had us continue to work with our 12 o’ clock partner. Once we had finished the problems on the work sheet we were to continue on the book work from yesterday. He had also given us the claims.

The claims for Sec 2-4 are numbers 13, 33e, 34d, and 41.

The claims for Sec 2-5 are numbers 13 and 20.

For the claims you need to submit the problems on loose leaf, in the form as though you were presenting it. Each problem will count as both a claim and a presentation grade. Have them finished by Tuesday were you will give the papers to Mr. Anderson to collect

Enjoy the half day tommorow.

Tommorow’s Scribee will be Nick Y. ENJOY