Last modified on August 21, 2012.
For each of the following items, use a graphing program, such as Graphing Calculator 3.5, GC 4.0, or GC Lite or other software, to explore, understand, and extend. 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 the problems for function grapher (y = f(x)) devices.
Investigate all of the problems. You may want to prepare a file of discussion, summary, or graphs to illustrate what you have found on some of the problems.
SELECT ONE PROBLEM TO WRITE UP AND POST TO YOUR WEB PAGE AS HOMEWORK. The write-up should deal with the underlying mathematics of the problem and not just be a collections of graphs from using the technology. The idea is to use the technology and what you glean from the exploration to communicate the mathematical ideas.
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? For example, what do you expect to change with a constant of 1 added to the right hand side of the equation?
Try several graphs in some systematic way. Click HERE for one set of graphs.
Try graphing this z = f(x,y), a 3-D surface. You may have to zoom in or out to get the image acceptable.
See Graph.
2. Make up linear functions f(x) and g(x).
Explore, with different pairs of f(x) and g(x) the graphs for
Summarize, explain, 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. Provide a mathematical interpretation of the Parameters a, b, and c. Explore using animations to illustrate the impact of each parameter.
6. Examine graphs of
for different values of a, b, and c. Graphing Calculator 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 or 4.0, graph
for various values of a, including a = 2, 3, e, 5, and 10. (Note: the prime notation will allow GC 3.5 or 4.0 to show graphs side-by-side.)
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 LemniscatesNote: 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.
