Mathematics Education

EMAT 6680, J. Wilson


EMAT 6680 Assignment 1




Last modified on August 24, 2009.


For each of the following items, use a graphing program, such as Graphing Calculator 3.5, or other software, to explore, understand, and extend. Prepare a file of discussion, summary, or graphs to illustrate what you have found. Some these explorations would be tedious with a function grapher (such as found on most hand-held graphing calculators) but you may want to consider the problems or modifications of them for function grapher (y = f(x)) devices.
Investigate all of the problems.

SELECT ONE PROBLEM TO WRITE UP AND POST TO YOUR WEB PAGE AS HOMEWORK.


1. Examine:

.

See the graph.What happens if the 4 is replaced by other numbers (not necessarily integers)? Try 5, 3, 2, 1, 1.1, 0.9, -3. Any unusual event? Interpret.

What equation would give the following graph:

What happens if a constant is added to one side of the equation? Try several graphs in some systematic way. Click here for one set of graphs.

Try graphing

See Graph.

Click HERE to open Graphing Calculator 3.5 to explore this equation.




2. Make up linear functions f(x) and g(x). Explore, with different pairs of f(x) and g(x) the graphs for

Summarize and illustrate.




3. Find two linear functions f(x) and g(x) such that their product

is tangent to each of f(x) and g(x) at two distinct points. Discuss and illustrate the method and the results.

 

Do you want to see someone else's discussion of this? If so click here.




4. Repeat Problem # 3 above where f(x) and g(x) are quadratic functions and each function, f(x) and g(x) is tangent to h(x) in two different points. That is, h(x) is a fourth degree equation and each of the second degree equations, f(x) and g(x) is tangent to h(x) in two points.






5. Examine graphs of

y = a sin(bx + c)

for different values of a, b, and c.


6. Examine graphs of

for different values of a, b, and c. Graphing Calculator 3.5 reserves the use of the symbol e.


7. Examine Graphing Calculator 3.5 graphs of

y = ln x

y = log x

Evaluate ln e and log 10. Also consider various values of a and b for

y = a (ln x)

y = ln (bx)

y = a (log x)

y = log (bx)


8. Using Graphing Calculator 3.5, graph


for various values of a, including a = 2, 3, e, 5, and 10.



9. Graph

What do you expect for the graph of

or




10. Let f(x) = a sin(bx + c) and g(x) = a cos(bx + c).

For selected values of a, b, and c, graph and explore:

 


11. Graph the equation

where |x| is the absolute value of x. Variations?




12. Explore the following equation for different values of a.

 



Some examples:

| a = 0 | | a = 1 | | a = 3 | | a = 5 | | a = 10 |

| a = -10 | | a = -3 | | a = -5 | | a = .10 |

| a = .50 | | a = -.50 | | a = -.10 |

| Multiple Graphs |




13. Consider two points (3,4) and (-5,-2). For any point (x,y) we can write the distance equations for these as


Explore graphs with these two distance equations. For example,

a. Consider when each is set to a non-zero constant. Circles are graphed.

b. Consider the sum

for various values of S.

c. Consider the product

for various values of P.


 

14. When the graph of the product forms two loops connected to a single point, a lemniscate is formed. If the two given points are (-a, 0) and (a,0) then the lemniscate has its center at the origin (0,0) and major axis along the x-axis.

For example, let a = 3. Then

 

will be this lemniscate:

a. Show that the equation can be simplified to

b. In general, if the foci of the lemniscate are (-a, 0) and (a, 0) then the equation in Cartesian coordinates is

Try graphs for different values of a.

c. Graph this equation

for different values of a and b.

 

d. Translate

into an equation in polar coordinates.



e. Other?

More about Lemniscates

Note: Later, we may have opportunity to examine lemniscates developed from geometric considertations. For example

-- a lemniscate can be formed by the envelope of circles with center moving along the hyperbola, all passing through a fixed point

-- a lemniscatge is image of the hyperbola transformed by inversion into a circle.


 

15. From Problem 13, consider

a. Graphs of the difference in the two distance functions being set to some constant. In particular, account for the possibility the that difference may be negative.

b. Graphs of the ratio of the two distance functions. In particular look at the ratio and its reciprocal.


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