Olive & Oppong: Transforming Mathematics with GSP 4, Page 1

Olive & Oppong: Transforming Mathematics with GSP 4, page 1

Chapter 7: Functions

As we saw in Chapter 6, dynamic dilations can be used to construct products and quotients of distances. This capability gives us a way of constructing algebraic relations geometrically with Sketchpad. Environments similar to Goldenberg’s Dynagraph (Goldenberg et al., 1992) can be constructed quite simply. The Dynagraph consists of two parallel number lines. The user controls the position of a variable point on one number line (the input variable “x”) and movement of this point causes movement of its image point (y) on the other number line according to a defined functional relation, y=f(x). With Sketchpad that functional relation can be constructed geometrically as well as typed in as an algebraic expression. Constructing algebraic relations geometrically can provide a powerful link between these two branches of mathematics and enhance learning through the dynamic exploration of functions.

Dynamic Function Representation on a Number Line

In the following section we shall be using our number line as a dynagraph to explore linear and quadratic functions. The input variable will always be a free point on the number line (labeled x) and the output of the function will be a point constructed from this free point using geometric transformations.

[Note: To construct a horizontal number line, define a coordinate system and then hide the grid and the vertical axis. Label the origin 0 and the unit point 1.]

Activity 7.1: Constructing a Linear Function

In order to construct a linear function on the number line you will need to create a free point on the number line for your “x” variable. Label this point x and find its x-coordinate. The next step is to create a segment that will represent the parameter a(or multiplier) of xin the function f(x)=ax. This could be done on the number line (as with the product A*B in chapter 6) but things start to get crowded and confusing if everything is on the same line. One solution is to create separate segments for each parameter in a function on hidden lines that are parallel to the number line. The following steps demonstrate how to create a segment for parameter aand construct the point ax on the number line:

1. Create a free point somewhere below your number line. Label this A.

[Note: You could place A on the vertical axis below the origin and then re-hide the vertical axis.]

2. Construct a line through this point parallel to your number line.

3. Mark the points 0 and 1 on your number line as a vector.

4. Translate your new point Aby the marked vector. This will create a unit point on your new line. Label it u.

5. Place a free point on your new line and label it a.

6. Hide your new line and construct the segment Aa.

7. Mark the DIRECTED RATIO Aa/Au. You will need to select your three points A, u andaIN THAT ORDER! then choose Mark Ratio under the Transform menu. [Note: Sketchpad always uses the following order when using 3 points to define a directed ratio: Common point, denominator point, numerator point.]

1. Mark the origin (0) of your number line as the center for dilation and dilate your point x by the marked ratio. Label the dilated image point ax.
2. Measure the abscissa (x-coordinate) of points x, A, a and ax.
3. Calculate the difference of the abscissas of points a and A(xa – xA) andlabel this measure “a”. Hide the abscissa measures xa and xA and edit their labels for xx and xax as in Figure 7.1.

Figure 7.1: Constructing the Function f(x)=ax.

Experiment by sliding your point aback and forth. What happens to x?What happens toax? Move your variable point x. What happens to ax? Does your point a change when you vary x?

Assignment 7.1

Construct a point on the number line representing the linear function f(x) = ax+b. Where b is represented by another segment constructed similarly to the construction of Aa. What kind of transformation of ax could represent adding a directed measure, b? Figure 7.2 illustrates one possible representation.

Figure 7.2: Constructing the Function f(x) = ax+b.

Experiment with different aand b. Vary x and observe the relative changes in the points ax and ax+b. Write down a conjecture concerning these two points. Figure 7.3 illustrates the situation for a negative value of b.

Figure 7.3: Showing the Relative Position of ax+b for Negative b.

Discussion. It is important to realize that you have built the function relation geometrically. What you observe is the dynamic effect of two transformations of a point. The algebraic expression ax+b is a mathematical way of representing the transformation of that point: a scalar multiplication (dilation) followed by a vector addition. What impact does the ability to directly vary the point x(and observe the effect on both points ax and ax+b) have on your concept of the function f(x)=ax+b? What might such experiences do for your students’ understanding of linear functions?

Constructing Powers of x

The problem of constructing a point that would be a distance x2 from the origin is a simple dilation of the point x by the directed ratio 0x/01. Create such a point on your number line. Use the default label x’ for this new point as Sketchpad has no superscript capability for its labels. Move your x point through the origin. What happens to x’x’? Does it behave as you would expect the point x2to behave?

Now construct a point that will be a distance x3 from the origin (use the same dilation on the point x’x’). Use the default label x’’x’’ for this point. Move your x point through the origin. What happens to x’’x’’? Does it behave as you would expect the point x3to behave?

The above process can be continued to construct any power of x.

Activity 7.2

Construct a point on your number line that can be represented by the quadratic expression ax2+bx+c where the parameters a, b, and c are defined by dynamic line segments (see Figure 7.4).

Figure 7.4. Constructing the Quadratic Expression ax’+bx+c.

Move your xpoint and try and predict the relative motions of each of the components of the quadratic expression (ax’, bx, ax’+bx, and ax’+bx+c). What happens to the different components as x passes through the unit point 1? What happens when it passes through 0? Can you find values (lengths) for b and c such that the point ax’+bx+c passes to the left of the origin (keeping a positive)?

Comparing Functions on Parallel Number Lines

Crowding all of the above points on one number line can become confusing. Keeping track of the relative motions becomes troublesome. Creating parallel number lines for each function or components of a function can help simplify the picture. One quick and easy way to create a parallel number line that maintains all of the functional relations is to translate your existing number line vertically.

1. Create a free point somewhere above the origin point. Label this point 0’.

[Note: You can show the hidden vertical axis and place 0’ on the vertical axis.]

2. Mark the vector 00’.

3. Select your number line and all the points on it.

4. Translate your number line by Marked Vector.

5. Translate this new number line by the same vector.

6. Repeat step 5 until you have one number line for each point.

1. Label one point on each number line and hide all of the other points on the line except your labeled point, the origin and unit point.
2. For each of your new number lines do the following in order to obtain a numbered axis:
3. Select the origin and unit point and choose construct circle from the Construct menu.
4. With this circle still selected, choose Define Unit Circle from the Graph menu. A dialog box will appear asking if you really want to create a new coordinate system. Click the Yes button.
5. Choose Hide Grid from the Graph menu.
6. Hide the vertical axis and the unit circle.

The above steps should leave you with something like Figure 7.5. Move the point x on the bottom number line and observe the relative motions of each of the points on the other number lines. Move xthrough the origin. Write down any conjectures you may have.

Figure 7.5: Multiple Number Lines to Represent Components of a Quadratic.

In order to investigate more closely the changes that occur as the x point passes through the origin you can “zoom in” on your number line by simply increasing the length of the unit segment 01. Figure 7.6 illustrates a zoom in to investigate changes between -1 and 2.

Figure 7.6: Zooming in on the Number Lines

Roots (or zeros) of quadratic functions can also be investigated on your parallel number lines. Simply move your x point and observe if or when the ax’+bx+cpoint passes through the origin for various values of a, b and c. The position(s) of your x point will be the zeros (or roots) of the quadratic function.

Assignment 7.2

Explore the roots for various values of a, b and c. Find values for which there are no roots, only one root, or two roots. Could there ever be more than two zeros for a quadratic function? Fix a and b and adjust c to create a function with just one root. Why does this work? Try adjusting a, b and c to find a function with a specific root. Figure 7.7 shows an apparent root at x = -2.

The multiple parallel number lines can also be used to investigate composition of functions. For instance, try creating points for the composition f(g(x)), where f(x)=ax2and g(x)=x-b. Explore this form of a quadratic. Where are the roots? Compare this form to the standard form. Describe the composition f(g(x)) in terms of geometric transformations.

Figure 7.7: Finding a Quadratic Function with a Root at -2.

Creating Dynagraphs Algebraically in GSP 4

The secret to constructing Dynagraphs algebraically using GSP 4 is to create two horizontal number lines (an input axis and an output axis) and to use the function calculator to calculate the value of your function for some variable point on the input axis. You then use this calculated value to plot a point on the output axis. As you move your variable point on the input axis the plotted point on the output axis moves appropriately. The major concern in using this method is to make sure the appropriate coordinate system is marked when you are calculating or plotting coordinates. Use the following steps as a guide:

1. Open a new sketch and choose Define Coordinate System from the Graph menu.
2. Place a point on the y-axis about an inch below the x-axis.
3. Hide the y-axis and hide the grid.
4. Select your point below the visible x-axis and choose Define Origin from the Graph menu. A warning dialog will ask you if you really want to define a new coordinate system. Click on the Yes button.
5. Hide the grid (choose Hide Grid from the Graph menu) and hide the new y-axis.
6. Label the top axis input and the bottom axis output (click on each axis with the label tool and edit each label).

At this point you should have two horizontal axes as in figure 7.8 below.

Figure 7.8: Two horizontal axes

1. Place a free point on the input axis and label it “x”.
2. VERY IMPORTANT STEP: Select the origin point of your input axis and choose Mark Coordinate System from the Graph menu (this step makes the input axis the active coordinate system for coordinate measurements).
3. Select your free point x on the input axis and measure its x-abscissa (from the Measure Menu).
4. Select New Function from the Graph menu and create your own function (e.g. f(x)=ex)
5. Select Calculate from the Measure menu. Click on your f(x) then click on your inputx value. Click on the OK button. The value of f(inputx) should be displayed.

At this point your sketch should look something like figure 7.9 below:

Figure 7.9: Input and function values

At this point there are two different ways to plot the output point on the output axis. You can select the origin of the output axis and mark it as the active coordinate system (see step 8 above) and then plot the point (f(inputx), 0) on this output axis. You will need a value 0 calculated or measured in order to do this. I prefer the following method, however, as it doesn’t require changing coordinate systems or creating a zero value. The following method uses the function value as a scale factor for dilation and then dilates the unit point of the output axis about the origin of the output axis using this scale factor:

1. Select the function value that you calculated in step 11 and then choose Mark Scale Factor from the Transform menu.
2. Double click on the origin point of your output axis. This selects it as a center of dilation.
3. Select the unit point on your output axis and then choose Dilate from the Transform menu. A dialog box should appear that indicates that you are to dilate by the scale factor of f(inputx). Click on the Dilate button. A new point should appear on your output axis. If you cannot see a point then move your input point, x, close to one or zero (depending on the function you used) until the point appears on your output axis.
4. Label this new point f(x). Construct a segment between points x and f(x).

At this point your sketch should look something like figure 7.10 below. This completes your construction of a Dynagraph. You can make a custom tool of this construction for creating more Dynagraphs. The givens for your dynagraph tool should be the origin point of a new input axis and a new function. (For tips on creating custom tools see your GSP manual.)

Figure 7.10: Completed Dynagraph in GSP 4

Challenge a classmate to try and discover your function by hiding the function expression and then ask your class mate to experiment by changing the position of the input point, x and observing the effect this has on the output point, f(x).

Exloring Range and Domain with Dynagraphs

Open the file Dynagraphs.gsp inside the Algebra folder. This sketch is from the CD accompanying Exploring Algebra with the Geometer’s Sketchpad (2003). You will find two pages of mystery dynagraphs. Explore each of them in turn and try and deduce the functions for each dynagraph. There are buttons to provide more information (such as the scale point – i.e. the unit point for each axis, or the actual numbers on the axes). There is also a button to show the functions but resist using this until you have thoroughly explored all of the dynagraphs on that page.

Comparing the range and domain of these different functions may help in your exploration. The domain of a function is the set of input values for which the function produces an output; the range of a function is the set of output values that the function can produce. In order to visually record the range of one of the dynagraphs in the Dynagraphs.gsp sketch, first select the triangular region attached to the output point and choose Trace Locus from the Display menu. Move the input point slowly to see the trace of the output triangle along the bottom of the output axis. Figure 7.11 shows the trace for functions h and i from page one of the sketch.

Figure 7.11: Two Dynagrpaphs with Ranges Traced

Note that the range for function h appears to be discrete rather than continuous, taking only certain values along the number line, whereas the range for function i appears to be only positive real numbers, but does appear to be continuous. What possibilities do these traces of the ranges of these functions suggest for the type of function in each case?

Explore the range and domain of the functions on page 2 of this sketch. Are there functions that have a limited domain (not all real numbers)? Are there functions that have a bounded range? Are there functions that appear to have “holes” in their range (a value or values that the function can not achieve)?

Activity 7.3: Composition of Functions using Dynagraphs

When we form the composition of two functions, such as g(f(x)), the output of the inner function (f(x)) becomes the input for the outer function. We can make this connection explicit using two dynagraphs. For instance, with the two dynagraphs in figure 7.11, in order to form the composed function i(h(x) we would want the output of h (the point h(c)) to be the input point for function i. The input point for function i is D. We need to make point D become point h(C). We can do this by splitting point D from its axis and then merging it with point h(C). The following steps achieve this process:

1. Deselect everything by clicking on a blank part of the sketch.
2. Select point D and then choose Split point from axis under the Edit menu. Point D (with its attached pentagon) will move away from the axis.
3. Leave point D selected and also select point h(C).
4. Choose Merge Points from the Edit menu.

Your two dynagraphs should now look like figure 7.12

Figure 7.12: A Composed Dynagraph

Now explore the range of this composed function. How is it related to the ranges of the two original functions? Describe the set of numbers that comprise this composed range.