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.