Mathematics Education

EMAT 6680, J. Wilson

EMAT 6680 Assignment 1

Last modified on July 11, 2008.

For each of the following items, use a graphing program, such as Graphing Calculator 3.5, or xFunction, to explore, understand, and extend. Prepare a file of discussion, summary, or graphs to illustrate what you have found.

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

i. h(x) = f(x) + g(x)

ii. h(x) = f(x).g(x)

iii. h(x) = f(x)/g(x)

iv. h(x) = f(g(x))

Summarize and illustrate.

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

h(x) = f(x).g(x)

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. Graph

What do you expect for the graph of

7. 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:

8. Graph the equation

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

9. 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 |

10. 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 C.

c. Consider the product

for various values of C.

d. 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:

Show that the equation can be simplified to

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.

Graph this equation

for different values of a and b.

e. Translate

into an equation in polar coordinates.

f. Other?

More about Lemniscates

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