Investigation #2 - Post by February 20th

For each of the following items, use a graphing program, such as Geometer's Sketchpad, Geogebra, or other software, to explore, understand, and extend. Prepare a page on your website of discussion, summary, and/or graphs to illustrate what you have found. Some these explorations are tedious if you rely solely on a hand-held graphing calculator. You will want to consider using a program that can graph implicit functions like maple or an online grapher 1 online grapher 2, online grapher 3. A 3D grapher (#1, #2) will also be useful.

Investigate all of the problems.

SELECT TWO PROBLEMS TO WRITE UP AND POST TO YOUR WEB PAGE.

1) Examine:

x(x² - 3) = y(y² - 1)

What happens if the 3 is replaced with other numbers (not necessarily integers)? First try replacing it with 2, 1, 0, -1, -2, then try some fractions and decimals. What do you notice?

Try graphing x(x² - z) = y(y² - 1)

*Hint: This graph is in 3d and you will need to solve of z.

2) Make up distinct linear functions for f(x) and g(x).

Explore what happens to h(x) for the following examples:

h(x)_s = f(x) + g(x)

h(x)_d= f(x) - g(x)

h(x)_p= f(x)(g(x))

h(x)_q = f(x)/g(x)

1) Can you generalize your observations? Are there any special cases?

2) What happens to h(x) when f(x) and g(x) are distinct quadratic functions?

Can you generalize your observations? Are there any special cases?

3) What happens to h(x) when f(x) is a linear function and g(x) is a quadratic function?

Can you generalize your observations? Are there any special cases?

3) Tangents

Find two linear functions f(x) and g(x) such that their product is tangent to both f(x) and g(x) at two distinct points. Discuss and provide examples used for the solution.

Are there multiple solutions to this question? Explain your reasoning.

4) Distance Formula

Consider the points (3,4) and (x,y). The distance between these points can be expressed using the distance equation and would look:

• Explore the graph of this function for different d values (real numbers). Interpret the graphs and discuss your findings. Use a 3d grapher to support any generalizations.

• Explore the similarities and difference if the point (3,4) is replaced with (1,2).

5) Cassini Ovals

The set of graphs that are made by the product of distances from two distinct points to an arbitrary point are called cassini ovals. An example of a cassini oval is:

Where the two distinct points are (4,0) and (-4,0). It's graph looks like this:

Explore the graph below for different values of p (real numbers).

What do you notice? Does your conjectures hold true for other cassini ovals for points (a,0) and (-a,0)?

6) Displaying data

Use geogebra to display data using each of these types of graphs:

1) Box plot

2) Histogram

3) Line of best fit

4) Bar Chart

(You will not be able to use one data set for all of the types of graphs. This means that you will have to choose multiple data sets and example why you can display the data using one of the 4 data displays listed above)

You can chose the data set from this webpage or another of interest. Discuss how you made each of the data displays and any difficulties you had while making them. Also, reflect on the value of using geogebra to present data in a k-12 classroom.

7) Circles.

Given two circles, there are 4 different ways they can be presented.

1) Totally disjoint

2) Intersecting at one point

3) Intersecting at two points

4) Intersecting at all points

Numbers 1 and 4 are trivial to produce so we will not focus on them for this investigation. Determine multiple ways of drawing one circle and then constructing the other circle in such a way that you produce #2 and #3. Consider circles of the same size and different sizes.

8) Cycloid

A cycloid is the path or locus of a circle as it moves along a straight line. The picture below shows such path.

Part 1) Construct a cycloid like the one pictured above and examine the path it makes.

Consider these questions while examining the path:

• How does the size of the circle effect the cycloid?

• What is the length of path of one cycle (between consecutive x-intercepts)?

Part 2) Cycloids can roll around other curves too. A cycloid that rolls around the inside of a larger circle is called a hypocycloid. Construct a hypocycloid like the one pictured below.

Investigate what happens when you change the size of the rolling circle.

• Can you make a hypocycloid connect to make a closed cycle? Compare the large and small circles when this happens.

• Will you observations be true if the circle rolls along the outside of the larger circle? (This is called an epicycloid)