Mathematics Education
.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.
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.
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.
