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

Learning Modules/Pre-Calculus/C.Akkoc/1996

MODULE No.2

EXPONENTIAL and LOGARITHMIC FUNCTIONS

Construct the graphs of the following exponential/logarithmic functions on TWIDDLE . Make a hard copy (3 pages) of each graph on a printer. Draw all asymptotes (vertical,horizontal,oblique) on the hard copy by hand (pencil only) using a ruler. Hand label (equations) all graphs and asymptotes. Graphs in the same group go on the same screen.

1. f(x) = 2^x; f(x) = 2^(-x); f(x) = 2^x + 2^(-x); f(x) = 2^x - 2^(-x)
   Twiddelish: f(x)=2^(a*x)+b*(2^(c*x)); Window: X[-5,5],Y[-5,5].
   
   
2. f(x) = log_200 (x) = (ln(x))/(ln(200))
   f(x) = log_10 (x) = (ln(x))/(ln(10)), common logarithms
   f(x) = log_e (x) = (ln(x)/(ln(e)) = ln(x), natural logarithms
   f(x) = log_2 (x) = (ln(x))/(ln(2))
   f(x) = log_1.5 (x) = (ln(x))/(ln(1.5))
   f(x) = log_1.2 (x) = (ln(x))/(ln(1.2))
   f(x) = log_0.9 (x) = (ln(x))/(ln(0.9))
   f(x) = log_0.5 (x) = (ln(x))/(ln(0.5))
   f(x) = log_0.1 (x) = (ln(x))/(ln(0.1))
   f(x) = log_0.005 (x) = (ln(x))/(ln(0.005));
   Twiddelish: f(x)=(log(x))/(log(a)); Window: X[-0.5,6],Y[-3,3].
   
   
3. f(x) = e^x
   f(x) = e^(-x)
   f(x) = ln[abs(x)]
   Twiddelish: f(x)=a*exp(x)+b*exp(-x)+c*log(abs(x)); Window:X[-5,5],Y[-5,5]

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

Learning Modules/Pre-Calculus/C.Akkoc/1996

MODULE No.3

GRAPHICAL COMPOSITION of FUNCTIONS

Given two functions f(x) and g(x). Calculate g[f(x)] and plot all three graphs [f(x),g(x),g(f(x))] on the same screen. Make a hard copy of the said graph on a printer (one sheet). Hand label all three graphs on the hard copy. TRACE the points x=-1/2,0,1 on the hard copy by drawing vertical/horizontal line segments with arrows to graphically simulate the composition x->f(x)->g[f(x)]. Then draw a vertical line segment connecting "x" to its direct image g[f(x)] on the graph. You should end up at exactly the same location you arrived after the two stage process described above. Mark and label all key points (coordinates) used in this simulation process on the hard copy.

f(x) = x^3 - 4x ; g(x) = 2x - 3

Twiddelish: f(x) = a * x^3 + b * x + c ; window: X[-3,+3],Y[-12,+6]

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

Learning Modules/Pre-Calculus/C.Akkoc/1996

MODULE No.4

GRAPHICAL INVERSION of FUNCTIONS

Given the function f(x). Calculate f^(-1)(x), the iverse function and plot both graphs [f(x),f^(-1)(x)] on the same screen. Make a hard copy of the said graph on a printer (one sheet). Hand label all graphs on the hard copy. Show f^(-1)[f(x)]=x to verify the inverse function found. TRACE the points x=-1,-2,-3 on the hard copy by drawing vertical/horizontal line segments with arrows to simulate the composition x->f(x)->f^(-1)[f(x)]. You should end up at "x", the same location you arrived after the two stage process described above. Mark and label all key points (coordinates) used in this simulation process on the hard copy. Draw by hand the line of symmetry y=x on the hard copy and mark pairs of points that are symmetric images of each other in f(x) and f^(-1)(x).

f(x) = (x + 3)^2 - 2 , x >_ -3

Twiddelish: f(x) = (x + a)^b - c ; window: X[-3,+5],Y[-3,+5].

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

Learning Modules/Pre-Calculus/C.Akkoc/1996

MODULE No.5

vertical/horizontal SHIFTS, REFLECTIONS, vertical/horizontal STRETCHING/COMPRESSION of FUNCTIONS

Given the function f(x)=x^2*e^(-x). Plot the following families of curves to demonstrate how the effects of various parameters lead to vertical/horizontal shifts, reflections, vertical/horizontal stretching/compression of functions. Make a hard copy of each screen (3 sheets). Hand label all graphs on the hard copy (equation). Mark the transformation (vertical/horizontal shift, reflection, vertical/horizontal stretching/compression) caused by a change in each parameter "a,b,c" on the graph of the genric function f(x), thus identifying the role of each parameter.

f(x)=x^2*e^(-x)

f(x) = a((b(x+c))^2*e^(-b(x+c)));
parametric form to demonstrate the effect of parameters [a,b,c] on various transformations.

Twiddelish: f(x)=a*((b*(x+c))^2*exp(-b*(x+c))); Window:X[-4,+4],Y[-4,+4]

Graphing Parameters:
b=1, c=0 ; a = -6, -3, 1, 6 -------> one screen
a=6, c=0 ; b = -2, -1, 1, 2 -------> one screen
a=-6, c=0 ; b = -2, -1, 1, 2 ------> one screen
a=6, b=2 ; c = -2, -1, 0, 1, 2, 3 --> one screen

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