Algebra Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Algebra/C.Akkoc/1996

MODULE No. 1

The PARABOLA: Horizontal/Vertical SHIFTS, Vertical STRETCHING/COMPRESSING, REFLECTIONS


  1. To enter the 2nd degree polynomial f(x)=a(x-h)^2+k, PARABOLA, into the graphing software " TWIDDLE " in generic form, type:

    f(x)=a*(x-b)^2+c

    where a=a, b=h, c=k in the standard form. Remember (h,k)=(b,c) are the coordinates of the VERTEX, while the sign of "a" controls the direction the parabola will open (a>0; opens upward, a<0; opens downward). Use a window with X[-15,10], Y[-10,10]. Draw the following graphs on the same screen.

    Pick a=1, b=0; draw parabolas for c = -8, 0, +5
    Pick a=0.5, c=-3; draw parabolas for b = -8, 0,+5
    Pick b=-8, c=+5 ; draw parabolas for a = -5, -1, -0.2, +0.1, +0.6

    Make a hard copy of your screen on the printer, write down the equation of every graph by hand on the hard copy. Put your name on the hard copy and turn it in.

  2. Given the basic function y=f(x)=(x^2)/2; draw graphs for the following functions on the same screen:

    (1) y=f(x), (2) y=f(x+5), (3) y=f(x-5), (4) y=6f(x-5), (5) y= -f(x)/2, (6) y=-3f((x-5)/4), (7) y=f((x+10)/3), (8) y= -f((x+13)/2), (9) y=-3f((x+10)/3)

    You first need to write/enter the given function in the generic form:

    f(x)=(a/2)*((x+b)/c)^2


    in order to create the functions in (1)-(9). These forms can be constructed by entering the appropriate values of the parameters "a,b,c" from the keyboard. Make a hard copy of your screen on the printer, write down the equation of every graph by hand on the hard copy. Put your name on the hard copy and turn it in.
On a separate sheet report your observations on the role each parameter [a,b,c] plays in generating the family of graphs. Which geometric characteristic(s) of the parabola do these three parameters control? That is; (1) flatness/sharpness, (2) location of the vertex, (3) the direction (upward/downward) the parabola opens.
 Algebra Module
Alabama School of Mathematics and Science
1255 Dauphin Street, Mobile, AL 36604

Learning Modules/Algebra/C.Akkoc/1996

MODULE No. 2

COMPARING the HARMONIC MEAN, GEOMETRIC MEAN, ARITHMETIC MEAN and the ROOT MEAN SQUARE of Two Positive Real Numbers


Given two POSITIVE REAL NUMBERS "a" and "b"; the following constitute a few ways the MEAN of these two numbers can be defined.


HM; the HARMONIC MEAN: 1/HM=(1/a+1/b)/2 ; HM=2ab/(a+b)

GM; the GEOMETRIC MEAN: GM=SQRT(ab)

AM; the ARITHMETIC MEAN: AM=(a+b)/2

RMS; ROOT MEAN SQUARE: RMS=SQRT[(a^2+b^2)/2]

The problem is to prove/show/demonstrate the following cascade of relationships between these different means for any pair (a,b):

2ab/(a+b)<_SQRT(ab)<_(a+b)/2<_SQRT[(a^2+b^2)/2]

To prepare for the said proofs first construct a TABLE displaying the following quantities for various combinations of (a,b). Using a spread sheet for this purpose will make life easier for you.

HM ______ GM ______ AM ______ RMS ______ HM*AM

INPUT (a,b): (0.1,100), (1,2), (1,3), (1,4), (1,10), (1,100), (1,1000), (2,3), (3,4), (9,10), (99,100), (999,1000).

Use 6 decimal place accuracy to get clues about how these means behave under different combinations of (a,b). Keep an eye for any trends that may be captured along the way. You may also experiment with additional number pairs on a spread sheet.


Analytical (algebraic):

  1. Prove the above inequalities using the property:

    for x>0, y>0; x <_ y <=> x^2 <_ y^2

Geometric:



Using the figure shown execute the following tasks to construct a geometric proof for the said inequalities:

  1. Explain geometrically why CE=EH=(a+b)/2=AM is true.
  2. Show that (geometric proof) DG=SQRT(ab)=GM and DH=SQRT[(a^2+b^2)/2]=RMS.
  3. Explain in geometric terms why DG<_EH<_DH and conclude from this that SQRT(ab)<_(a+b)/2<_SQRT[(a^2+b^2)/2].
  4. What happens to these inequalities for the special case where a=b? Please calculate and explain the geometric significance.
  5. Try to construct a GEOMETRIC INTERPRETATION for the Harmonic Mean on the Figure shown above, thus confirming the algebraic proof in Part1 (algebraic). That is, can you find a geometric CONSTRUCTION where a line segment will be equal to the harmonic mean ?