Today we studies the Double Angle ID's: sin(2A)=2sinAcosA and cos(2A)=cos^2A-sin^2A. Once you solve for both sin(2A) and cos(2A), you can solve for tan(2A)=sin(2A)/cos(2A). It's important to know that for that the 2 in 2 sinAcosA isn't the same as the 2 in its equivalent sin(2A). Its derived from the sum of the 2A represented in the equation 2(sinAcosA).

12/2/09 Sum/Difference Identities

In class on December 2nd we learned about the Sum and Difference identities. (Note: a large portion of these notes show how we derived the identities given angles and prior knowledge of the unit circle.)

The identities allow us to find the sine, cosine, and tangent of angles we previously could not (ie: 15 degrees), by adding or subtracting (ie: 60-45=15) using angles for which we can already find the trig functions.

Coefficients of Angles (Trigonometry)

We discussed in class that normally when using trig identities to the meaasure of an angle, you come out with two possible angles. However, a coefficient of an angle produces that double that number of solutions.
girl in the hall: "It took me three minutes to get ready today"
MV: "I can tell"

Notes: 11/30

Sorry it took me this long to post the notes. I had a few problems. Anyway, today in class we learned how to solve trig equations by using 3 main steps:
1.Get the 1st answer from the inverse trig function
2.Then the second answer from the reflection of the trig ID (Ex: Tan= 180 or 2pie+the angle)
3.Add or subtract 2 pie or 360 (depending on whether you are in degrees or radians) to both of your answers from 1 and 2 to get as many answers as needed.

11/23 Notes

In class, we worked with trigonometry identities. When you have a trig equation, you want to first look to do algebra: factor, expand, or get a common denominator. Second, look for trig IDs to use. We learned definitions ie sec(x)=1/(cos(x)). We also looked at the pythagorean IDs: cos^2(x) + sin^2(x)=1, 1+tan^2(x)=sec^2(x), cot^2(x)+1=csc^2(x). We also studied complement IDs (ex. cos(x)=sin((pi/2)-x))and reflection IDs (ex. sin(x)=(pi-x)). We applied these IDs to simplify equations and expressions.

11/17 class notes

Today in class we talked about inverse trigonometric functions, like arcsin etc. The main concept we covered was the ambiguity of the number of solutions for these functions. In other words, how do we restrict arcsin etc. to only produce one solution? (for example if you plugged in arcsin= .5 to your calculator how could you insure that it would give you the answer 30 degrees rather than 390, - 330, etc?) Well what we do is restrict the domain and range. In the chart in the notes it lists the different domain and range restrictions for the different functions. The reason we have to insure they only have one solution is that if they had multiple they would not be functions anymore (aka more than one output).

These notes are about graphing tan

November 9 - Trig Graphs

Today in class we started by reviewing the basic facts about the graphs of y = a cos (bx) and y = a sin(bx). We moved on to an example of writing an equation for a given trig graph. You need to consider how to find the amplitude (a = (Max - min) / 2), the vertical shift (k = (Max + min) / 2), and b by recognizing the period (period = 2π / b). Once those values are determined, consider the graph of y = a cos (bx) + k. Finally, determine how that graph can be shifted to "overlap" the given graph to obtain a final equation of the form y = a cos(b(x - h)) + k.

The same process can be used to obtain an equation of the form y = a sin (b(x - h)) + k. Note, there is not one unique answer in any case.

Sine and Cosine Curves

First off, I would like to tell everyone I'm sorry I didnt get this on here sooner if y'all needed. Anyway, Friday in class we discussed the graphs of both sine and cosine graphs. We covered the topics of period and amplitude as well as the standard form of an equation for said graphs.

Today we continued our discussion on trig functions, however we incorporated reflections and the Pythagorean theorem. We learned how to find the sine, cosine, and tangent of theta when it is reflected across the x-axis, y-axis, or origin. We used the Pythagorean theorem to find the missing side of a triangle and the equation of a circle (see HW notes).

Notes, 11/3/09

Main Topics of Lesson...

• unit circles
• the coordinate (cos of theta, sine of theta)
• the four other trigonometric functions
• reflecting theta across the axes and origin

Notes follow...

Notes, 11/2/09

Today we began to move into trigonometry by looking at angles and their measures. The two units of measurements for angles are degrees and radians. It should be noted that one radian is the measure of the angle formed at the center of a circle when the arc length is the same as the radius. This comes out to approximately 57.1 degrees. (see notes for picture) We also looked at how to determine arc length, as well as supplementary and complementary angles. Notes below.

Welcome (back) to the PreCal Blog

So I have finally sent out author invitations. In class on Monday, we will talk about responding and signing up and your once-every-forty-two-days responsibility of posting to the blog.

September 17/18 - Function Transformations

We spent time in class on each of Thursday and Friday talking about the more challenging function graphs that I posed on Wednesday. Those graphs were:

1. y = v(|x|)
2. y = v([|x|])
3. y = [|v(x)|]
4. y = v(1/x)
   

The main idea I want students to get from this activity is that you can always generate points on the "new" graph you are trying to draw by using appropriate points from the "old" (original) graph. The challenge is often in deciding which points to use and how many points to use.

Rather than posting my notes from class here, I have posted a file on my website that contains images of the graphs above.

On Friday in class, I handed out a copy of the first test from last year. Your test on Tuesday will be similarly constructed.

September 16 - Function Transformations (Challenging)

Yesterday (Tuesday, Sept 15) and today, we have been reviewing the basic transformations (up, down, left, and right shifts, reflections, stretches and shrinks). In the homework, Tuesday night, I introduced some function composition (using x^2, |x|, 1/x, and Sqrt(x)).

Today in class, after reviewing the homework, students were put in groups and given a single graph to work on. Tomorrow in class, each group will present their graph and explain how they came up with it, hopefully convincing the rest of the class that they are correct.

This is a challenging exercise. I hope that you start to understand how to generate a graph of this sort, but it may take some time and practice.

There are no notes for yesterday or today.

September 14 - Function Transformations

We started class today with a clip from comedian Brian Regan about calling up UPS to pick up some boxes. This may or may not help you remember the definition of girth.

We worked a couple problems from Friday's homework. I hope that students see the details involved in describing the inverse of a given function, especially with regard to domain issues.

I presented one six-part question as a review of the ideas of function transformations. Here it is:

Given (4, 1) is a point on the graph of y = f(x), determine a point on each of the following graphs: (note the answers are in red...)

1. y = f(x) + 6 (graph moves up 6, so (4, 7))
   
2. y = f(x – 6) (graph moves right 6, so (10, 7))
   
3. y = f(6x) (horizontal shrink, so (2/3, 1))
   
4. y= 6f(x) (vertical stretch, so (4, 6))
   
5. y = Sqrt(f(x)) (similar to #4, only the y value is affected (and since Sqrt(1) = 1), the answer is (4, 1))
   
6. y = f(Sqrt(x)) (similar to #3, only the x value is affected, so the answer is (16, 1))
   

September 11 - Inverse functions

The topic for today was inverse functions.

The inverse of a given function, f(x), is derived by taking the order pairs that make up f(x) and switching the inputs and outputs. So if (a, b) is a point on f(x), then (b, a) is a point on the inverse of f(x).

Note the inverse of a function might not be a function. No biggie.

(notes for Friday)

Sept 10 - Function Operations (including Composition)

We had a quiz in class today. The bonus was a bit of a fiasco for my 6th period class - but I have a solution for that - I'll talk about it in class tomorrow.

Function composition is simply the operation of substituting one function into another one. If we have two functions f(x) and g(x), I can write f(g(x)) ("f of g of x") or g(f(x)) ("g of f of x"). Generally speaking, they will not be the same (function composition is not commutative). A couple of the problems may be tricky because you have to determine the output of the function from a graph, not a formula.

My notes for today are brief.

Sept 9 - Functions with Restricted Domains

Today's topic is a favorite of mine. I like these types of problems for where we are in class because they draw on a number of different topics. The main concept is graphing a function (without using a calculator) by using knowledge of parent functions and transformations. Then you'll have to analyze the graph to determine the range.

I'm posting the examples from class. The quiz tomorrow will be a couple of problems like these examples or like the homework problems.

Sept 8 - Function Transformations

Not a lot new today. We finished the game from Friday and I reviewed the rules for function transformations. Basically, we are now working toward generating graphs of functions without using a calculator.

I am posting the 12 functions that I graphed in class Fri and today.

September 4 - Parent functions

We played a graphing game in class today. I projected the graph of a function on the board, and then groups in turn would guess what they thought the function was. The first round was pretty easy, while the second round was problematic as many students forgot about 1 / x and the greatest integer function. The third round was the challenge round as I changed the scale of the graphs but did not tell what that scale change was. We will complete the third round in class on Tuesday.

I have included the list of parent functions for which you are responsible below. I have also reminded you how the graphs shift right/left and up/down depending upon h, k in y = f(x–h) + k.

September 3 - Quiz

We had a quiz in class today. I anticipated that students might take longer than most did - I think that means that you are feeling very confident about the domain/range topics (and using the calculator to generate graphs).

The homework tonight is just an algebra assignment, solving for a variable in one equation and substituting into the other. However, each of these problems came from some other more complicated (calculus) problem, so if it seems easy to you - that means you're understanding a big part of the process of solving some calculus problem. Cool!

There were no notes today.

September 2 - Piecewise, Increasing/Decreasing

Two different topics today.

First, I reviewed piecewise functions, specifically how they are graphed. For now we will use the calculator to graph functions. Eventually, we will be able to work with piecewise functions without relying on the calculator.

The other idea was describing when a function is increasing, decreasing or constant. Remember, that when you determine the interval on which a function is increasing, you are describing x values. Also, the convention is always to use open intervals to describe increasing or decreasing behavior.

September 1 - Domain and Range

Determining the domain of a function is fairly straightforward. There are a couple things to consider - division by zero and the presence of negative numbers in an even root.

Determining the range of a function is a little more involved. For me, range is a visual thing and so I encourage you to graph functions. As we go on this semester, there will be more functions you can graph on your own and fewer functions for which a calculator will be necessary. For now (and for the quiz on Thursday), the expectation will be if I ask you for range, you will either graph the function on your calculator or I will give you the graph.

August 31 - Functions

We spent most of today going over the homework from Thursday night - drawing graphs to represent different real world functions.

I mentioned to my sixth period class something that I should have told my 1st and 3rd period classes - while you were presented a lot of information in class today, I'm not going to hold you fully responsible for it. If you forget for now that I can model how a rumor spreads with a "logistics model," that's fine. My main goal for today was to impress upon you the variety of functions that are necessary to model real world behavior.

The main concepts for today were the definition of a function and how function notation works.

August 28 - Intro to Functions

I returned the quizzes from yesterday; overall they were very good.

We played my "functions" game in class today. I like to emphasize that functions don't have to be restricted to the generic f(x) = x^2 or whatever that you are used to. The idea of an input/output relation is very broad and math is applicable to the real world because clever mathematicians find ways to apply the generic f(x) in clever ways.

There is no assignment for the weekend. We will talk about the graphs from the functions worksheet (Thursday night's hwk) in class on Monday.

August 27 - Quiz over quadratics/Functions

Most of class today was spent taking our first quiz of the year.

The homework for tomorrow is for students to draw graphs for each of 15 different functions that I have described. Many of the graphs will be fairly general, but in class, we will eventually talk about what kind of function (linear, polynomial, exponential, trigonometric, etc.) would describe the scenario best. I really like this assignment as a jumping off point for talking about functions in general.

There are no notes for me to post from today.

August 26 - Quadratics

I announced a quiz for tomorrow (Thursday) today in class. It should be a fairly short quiz over quadratic functions and calculator use will be allowed. I explained that typically, quizzes are untimed - students can take as long as they wish to complete the quiz (within the class period). I will remind you that after you turn your quiz in, you need to try and stay quiet so that others can finish their quizzes without too much distraction.

I worked an example of a somewhat tricky quadratic word problem. Note that I posted worked out solutions to the worksheet on my website.

August 25 - Quadratic Functions

We started by going over several of the homework problems from last night.

The topic for today is quadratic functions. Like yesterday with linear functions, I'm not sure that there is anything here that is new or particularly difficult. I walked through the use of the calculator for finding the equation of a parabola given three points (using the QuadReg feature). Other than that, students seemed very comfortable with today's concepts.

The one tricky thing I did was to write a problem with the x's and y's switched. I think it's good practice to be able to work problems with vertically or horizontally oriented parabolas.

Here are my notes for today.

August 24 - Linear Functions

We started by going over a couple problems from Friday's assignment over inequalities.

Today's main concept is linear functions. There is really nothing complicated about lines or linear functions. Having said that, the idea of slope (especially that slope represents rate of change) is in fact a hugely important concept, and it will eventually be one of the building blocks of calculus.

I announced in class that there will be a quiz on Thursday (covering quadratics, which we will begin tomorrow).

Here are my notes from today.

August 21 - Solving Inequalities

We reviewed a couple problems from the homework last night. Problem #151 is tricky because there are extraneous solutions - it is necessary to check your answers to determine which are the "true" solutions. I pointed out that #134 should not have been assigned b/c we did not cover solving equations with radicals in them in class on Thursday.

In class, we covered solving inequalities. The types of inequalities you are responsible for are simple linear or compound linear, absolute value, and rational. When an inequality is rational (or polynomial), the strategy for solving will be to set one side of the inequality equal to zero and then to mark the zeros of all factors (in the numerator and the denominator) on a number line. Then test intervals to determine where the expression is positive and negative.

August 20 - Solving Equations

Today in class, we reviewed solving several types of equations.

These types included linear, quadratic, quadratic-ish, absolute value and radical equations. We also talked about factoring by grouping and working with rational expressions.

Note that in all cases, the problem eventually broke down to solving a linear equation (or at least, solving several linear equations)