Function+Transformations


 * Function Transformations **


 * Sample Scenario 1 :** I have a square room. I am going to cover the floor with 1m x 1m red tiles - everywhere but in the four corners. In each of the these corners I want to place a 1m x 1m blue tile . If the side of my room is //x // m, how many blue tiles do I need? How many red tiles do I need?


 * Sample Scenario 2 :** I have a square room. I am going to cover the floor with 1m x 1m red tiles - everywhere but along the walls. Along each of the walls I will place 1m x 1m blue tiles . If the side of my room is //x // m, how many red tiles do I need?


 * Sample Scenario 3 **: A carrier pigeon flies directly east or west from MiddleLand She charges me $3 flat fee plus $0.50 for every mile she flies. If MiddleLand is at (0,0) and the x-axis is distance in miles east-west and the y-axis is payment in $, find a function that describes what I have to pay the pigeon.


 * Sample Scenario 4 **: On my first day of work at the laboratory I was given a sample of radioactive material that weighed 15g and told that the half-life of this material was 15 days. Find the function that describes how much of the material remains as a function of days. (A question: I know that uranium decays to lead. Does it do so evenly? That is, is the lead mixed in with the uranium? If so, how did they find the half-life?)


 * Other scenario ideas:** Interest, Free fall with no initial velocity.

Metadata CCF-BF-3

**Prerequisite knowledge:** The graphs of the //**core functions**// you plan to transform together with their key points. Probably:

Presentation Ideas: I would probably focus on and because these functions are defined everywhere and is neither even nor odd. Immediately after doing a couple of standard single transforms with the rules, I would start using key points.

Then, I would do all of the above transforms for -f(x) and f(-x) since these are the transforms used most frequently in life and even in mathematics.

At the end, I would want them to **work backwards**, e.g. graph sqrt(x+1), 3sin(x), y=ln(-x) and ask them to find the function transform. Further. Find a function transform of sin(x) that is shifted to the right 2. Find a function transform of sin(x) with a period of 10. Find a function transform of sin(x) with an amplitude of 5. Put them together. (Gets them ready for sinusoids.)

a. First explain "the rules" for shifts and scaling
 * f(x)+3 moves f(x) up 3, f(x)-2 moves f(x) down 2.
 * f(x+3) moves f(x) left 3, f(x-2) moves f(x) right 2.
 * 3f(x) stretches f(x) vertically by 3, f(x)/2 flattens f(x) vertically by 2.
 * f(3x) compresses f(x) horizontally by 3, f(x/2) stretches f(x) horizontally by 2.

b. That is, "outside" transforms do what we think, but "inside" transforms do the opposite of what we think. For example f(x)+3 moves f(x) **up** 3, but f(x+3) moves f(x) **left** 3. Here is how I think about this. When doing function transforms, I use "key points" of f(x). Suppose **x=1 is a key value of x**. Then, **x=1** must go **into** the function machine.
 * f(x)+3 changes the value coming out of the function machine.
 * f(x+3) changes the value **going into** the function machine.
 * f(x)+3. Since transform is outside, changes value "after", so I test x=1.
 * f(x+3). Since transform is inside, changes value "before" so I need to find x so that x+3=1 => x=-2 so I test x=-2.

c. y=x-2 can be written y=(x)-2 or y=(x-2). Graph both of these as a transform of the function y=x. Did you get the same graph?
 * Good Questions:**

Resources: Typical, but well explained: http://www.youtube.com/watch?v=3Q5Sy034fok