FSTChapter+2

toc =**Chapter 2: Functions and Models**=

**2-1: The Language of Functions**
The independent variable determines what happens with the dependent variable. You can always figure this out by thinking about what is going on in the situation you are looking at. - Zachary and Jeremy

A function is a set of ordered pairs (x,y) in which each value of x is paired with exactly one value of y. In the set of ordered pairs of a function, the set of first elements is the domain of the function. The set of second elements is the range. Another definition for function is a correspondance between two sets A and B in which each element of A corresponds to exactly one element of B. In this form the domain of the set is set A, and the range is the set of only the elements of B that correspond to elements in A. Functions can be shown in many ways, including ordered pairs in tables or lists, rules expressed in word or symbols, and coordinate graphs. A way to check if a set is a function is the vertical line test. First you graph the set and draw vertical lines on the graph they should only intersect one point. If it intersects more then one, the set is not a function.

A-C Football is going to win

**2-2: Linear Models and Correlation**
A **__linear function__** is a set of ordered pairs in the form of y=mx+b. You can do this on your calculator by entering all your ordered pairs into L1 and L2. then press STAT, go to CALC, and press 4, which is LinReg(ax+b). Now, back on the main screen, you need to tell the calculator where you want to get your dependent and independent variable, and where you want the line of best fit. So right now all you should have is LinReg(ax+b). Press 2ND 1 then the COMMA button (above #7) then press 2ND 2 and COMMA again then press VARS, then Y-VARS, press 1 for Function, then 1 for Y1. This will put your line of best fit in Y1 under the Y=. Just press enter and BAMM! you got a graph. With this, you can find __**interpolations**__, values between know values on the graph, and **__extrapolations__**, values beyond the graph. It can also help you find out if the graph has a **__strong correlation__** or a __**weak correlation**__. A strong correlation will have points very close to eachother following the line of best fit. A weak correlation will have point all over the place and it will be hard to tell which way they are going. Hope this helps!

-Clarky and Mock Daddy

**2-3: The Line of Best Fit**
The line of best fit is a line, when the the value of the sum of errors squared is the smallest. You can find this on the calculator and by hand. Simply use the steps clarky and mock talked about in lesson 2-2 for the calculator, or simply set up a system to do it by hand. To help you find the line of best fit, you will need to understand a few terms. The observed values are the values already found in your initial data, and the predicted values are the dependent numbers found from your line of best fit. Finally you can use these two to find the errors(residuals) its as easy as subtracting the predicted values from the observed values. Square the errors, and the smallest sum is the best choice for the line of best fit. Troy L. and Ally R.

**2-4: Exponential Functions**
This section re-introduces the idea of the exponential function. Exponential functions are of the form //y// = //ab//^//x. a// will never have a value of 0, as all output values would also have a value of 0. There are two situations for //b//. First, when //b// > 1, we will have an exponential growth situation. Here, we will be multiplying by a percentage greater than 100%, thus resulting in a growth. The graph will start near the //x//-axis (but will never touch it, making it an asymptote), and it will rise at a quicker and quicker rate as //x// increases. This is a strictly increasing situation. As the independent variables increase, the dependent variables also increase.

The second situation is when 0 < //b// < 1. Here we will end up multiplying by a value between 0% and 100%. Since it is not the entire amount that we are multiplying by, we will end up with less than what we began with. The graph will start out very high and get closer and closer to the //x//-axis without ever touching it. This is a strictly decreasing situation. As the independent variables increase, the dependent variables decrease.

In both cases, we will have a //y//-intercept of (0, //a//).

-Mr. Lamb



**2-6: Quadratic Models**
Quadratic models are situations where an equation of the form //y// = a//x//^2 + b//x// + c will describe the trend in the data. These models will give us graphs that are parabolas. These graphs open up when a > 0 and down when a < 0.

The minimum or maximum point will occur at the vertex, which can be found by using the formula //x// = -b/2a. Once you find the //x//-coordinate, you can find the //y//-coordinate by plugging //x// into your equation. = =

One very important use of the quadratic model is with Newton's Formula //h// = -1/2 //gt//^2 + //v//0//t + h//0, which models the path an object takes while in motion. //g// is acceleration due to gravity, which is either 9.8 m/sec^2 or 32 ft/sec^2, depending on what units you are using in your problem.

You can find an exact quadratic model for any three points, but that model often does not fit the entire set of data. To find the best fit, you want to use the quadratic regression in your calculator, which takes into account all of your given data points.

Sometimes you will find a model that doesn't really follow any theory to explain why the model fits. This is known as an impressionistic model.

-Mr. Lamb

**2-7: Step Functions**
Step functions are basically defined in their name- functions whose graphs resemble steps. Because there are steps, the graph is termed //discontinuous//. This means that this function cannot be graphed without lifting your pencil. In a discontinuous graph, there are //points of discontinuity//, or places where the step jumps to the next step. Unlike a discontinuous graph, a continuous graph has no holes or jumps. There are two types of step functions: ceiling and floor functions. Ceiling functions are also termed //rounding-up function//s which are also defined in their name, and round non-integers to the nearest integer greater than it. Floor functions are also termed //rounding-down functions// quite appropriately because they round non-integers down to the nearest integer less then it. Notating for ceiling and floor functions is quite simple. For a floor function, the symbols l_ _l (well, resembling it at least) must be used. For instance, l_ 4.8 _l = 4. For a ceiling function, the floor function symbols must be flipped upside-down. If one would have to graph one of these functions, one would have to realize that (a) the graph would resemble steps (as stated before) and (b) one of the endpoints of the graph would be a filled-in dot, indicating that the point is included in the graph and one of the endpoints would resemble a tiny circle, indicating that the endpoint is //not// included in the graph.

**2-8: Choosing a Good Model**
In this chapter, the topic that was discussed was how to choose a good model. In order to find a line of best fit, you can't always depend on a strong correlation coefficient (r). When you graph the residuals, the closer the points are to the x-axis, the better the model. The points, however, should not show a trend- for instance a parabola or an exponential curve.
 * **__Residuals-__** is a synonym for error that is often used in statistics; error = observed //y// - predicted //y//

By Booooohn and Z. Gibson

**2-9: The Men's Mile Record****
This lesson talks about the men's mile record. It talks about fitting a model to that data. They show the models for linear, exponential, and quadratic. If you look at the models you would find out that exponetnial would fit the best because linear and quadratic would keep going below the x-axis and that is impossible. The lesson also suggests that you could study the average speed of a runner rather then how long it takes to run a certain difference. If you do the work like the book did, you find out that you will get around the same answer as the other way. The mile record as of 1996 was 3:44.39 set by Noureddine Morceli of Algeria in 1993.

Travis D