Goal:

Understand what is a linear function, and the representation y=m+k(x-a) of an affine (constant + linear) function. Recall the characteristic features of the monomials f(x)=x^m with m a natural number, and what is a polynomial. Study in some detail quadratic polynomials, and compositions (sammansättningar) with affine functions.

Program:

1. Start matlab, add in your setpath.m file
addpath 'L:\Matkfkb\Kfkb\guis\ascii'
addpath 'L:\matkfkb\KfKb\guis\binaries'
and save and run the script to get the corresponding additions to your matlab search path.

2. Give the command open('RM+.fig') (or possibly open('RMplus') if this doen't work) to open the Road Map to the Mathematics Laboratory. Enter the Graph Gallery, and then the exhibition of linear function by pressing the linears button. Alternatively you may enter this lab directly from RM+ by pressing the Linears button there, or directly from the matlab prompt by the command open('GGlin.fig').

3. Start by choosing f(x)=m and changing m with the m-sliders. Then proceed by choosing f(x)=m+kx and changing m and k. Note the way the graph and the equation changes.

4. Create a linear function with a) f(0)=1 and f(2)=0, b) f(0)=-.2 and slope = .6, c) f(-1)=0 and f(1)=1

5. Proceed by choosing y=m+k(x-a) and changing m, k and a.
Make absolutely sure you fully understand the role of each of the parameters m, a and k in this relation, which may well be the most important you will ever meet.

6. Create linear (affine) functions with a) f(1)=1 and slope -.2, b) f(-1)=0 and f(1)=1.

7. Press the questions button and answer the listed questions.


8. Now enter the polynomials gallery and lab, from the Graph gallery of directly from the RM+ map.

9. First study monomials of the type x^m. What is the significance of a) odd/even m values of m, b) the size of m, c) the case m=0.

10. Recall that a function f is called odd if f(-x)=-f(x) for all x, and even if f(-x)=f(x) for all x. Are some of the functions x^m odd/even? Is the function x^3-2x odd? Is the function x^2-x^3 odd or even?

11. Study the role of a and k in the relations y=(x-a)^m and y=k(x-a)^m, respectively.

12. Study polynomials by (from RM+) pressing the polynomials button and selecting f(x)=k_0+k_1x+k_2x^2+.. and vary the multiples k_m by, for each selected m value, setting k_m with the k-slider or edit box.

13. Create a polynomial of degree 2 with f(0)=0 and maximum = 1 for x=.4

14. Create a polynomial of degree 3 with (approximately) a local min value = -1 for x=1 and a local max value = 1 for x=-.4, by trial and error.

15. Create a polynomial function of lowest possible degree with two (local maxima) and one (local) minima.

16. Invent your own problems and experiment!!

17. Reproduce the current plot in a new window by commands directly from the matlab prompt by first typing figure, to get a new window, then defining an appropriate range of x-values like x=0:0.01:1, then the corresponding y-values, for example y=1+2*x-3*x.^2, and finally typing plot(x,y)

18. Check that you have reached the goals of the studio session!

/Nils