Olive Oppong Transforming Mathematics with GSP 4 page 76 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 xcoordinate The next step is to create a segment that will represent the parameter a or multiplier of x in 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 Olive Oppong Transforming Mathematics with GSP 4 page 77 hidden lines that are parallel to the number line The following steps demonstrate how to create a segment for parameter a and 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 A by 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 and a IN 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 8 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 9 Measure the abscissa x coordinate of points x A a and ax 10 Calculate the difference of the abscissas of points a and A xa xA and label this measure a Hide the abscissa measures xa and xA and edit the labels for xx and xax as in Figure 7 1 x 2 01 ax 3 55 ax 10 5 0 x A 1 u 5 a a 1 76 Figure 7 1 Constructing the Function f x ax 10 Olive Oppong Transforming Mathematics with GSP 4 page 78 Experiment by sliding your point a back and forth What happens to x What happens to ax 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 x 2 01 ax 4 04 ax b 1 95 ax 10 ax b 0 x 5 A 1 u 5 10 a a 2 01 B u b b 6 00 Figure 7 2 Constructing the Function f x ax b Experiment with different a and 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 x 1 98 ax 3 99 ax b 2 01 ax ax b 10 0 5 A b 1 u x 5 a a 2 01 B u b 6 00 Figure 7 3 Showing the Relative Position of ax b for Negative b 10 Olive Oppong Transforming Mathematics with GSP 4 page 79 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 Does it behave as you would expect the point x2 to behave Now construct a point that will be a distance x3 from the origin use the same dilation on the point x Use the default label x for this point Move your x point through the origin What happens to x Does it behave as you would expect the point x3 to 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 Olive Oppong Transforming Mathematics with GSP 4 page 80 x 2 02 ax bx x bx 10 0 5 A b 1 x ax ax bx c 5 10 ua …