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
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.]
Olive & Oppong: Transforming Mathematics with GSP 4, page 77
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
10
a
a = 1.76
Figure 7.1: Constructing the Function f(x)=ax.
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.
Olive & Oppong: Transforming Mathematics with GSP 4, page 78
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
10
a
a = 2.01
B u
b = -6.00
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
Olive & Oppong: Transforming Mathematics with GSP 4, page 79
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).
x = 2.02
ax'+bx x'
bx
-10
0
-5
1
A
x
ax' ax'+bx+c
5
10
ua
a = 1.41
b
B
u
b = -1.38
C
u
c
c = 4.48
Figure 7.4. Constructing the Quadratic Expression ax’+bx+c.
Move your x point 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
Olive & Oppong: Transforming Mathematics with GSP 4, page 80
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.
7. 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.
8. For each of your new number lines do the following in order to obtain a numbered
axis:
a. Select the origin and unit point and choose construct circle from the
Construct menu.
b. 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.
c. Choose Hide Grid from the Graph menu.
d. 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 x through the origin. Write down any conjectures you may have.
Olive & Oppong: Transforming Mathematics with GSP 4, page 81
ax'+bx+c
-10
-5
0
-10
-5
0
-10
-10
-5
bx
10
ax'
5
10
0
5
10
5
10
5
10
-10
-5
0
-10
-5
0
A
b
10
5
0
-5
5
ax'+bx
B
x'
x
1
ua
x = 2.02
a = 1.41
u
b = -1.38
u
C
c
c = 4.48
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.
ax'+bx+c
0
-1
0
-1
1
2
1
2
ax'+bx
ax'
0
-1
1
2
x'
0
1
2
-1
0
1
2
-1
0
-1
bx
A
11
a
x = 1.37
u
a = 0.24
b
u
B
b = -0.23
c
C
u
c = 0.75
Figure 7.6: Zooming in on the Number Lines
x
2
Olive & Oppong: Transforming Mathematics with GSP 4, page 82
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+c point 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)=ax2 and
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.
ax'+bx+c
-10
-5
0
5
-10
-5
0
5
-10
-10
-10
-5
-10
-5
0
x = -2.00
5
10
5
10
5
10
5
10
bx
0
x
10
x'
0
-5
1
A
ua
a = 1.41
b
B
u
b = -1.23
c
C
10
ax'
0
-5
ax'+bx
u
c = -8.11
Figure 7.7: Finding a Quadratic Function with a Root at -2.
Olive & Oppong: Transforming Mathematics with GSP 4, page 83
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.
input
-10
-5
5
10
5
10
output
-10
-5
Figure 7.8: Two horizontal axes
7. Place a free point on the input axis and label it “x”.
8. 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).
Olive & Oppong: Transforming Mathematics with GSP 4, page 84
9. Select your free point x on the input axis and measure its x-abscissa (from the
Measure Menu).
10. Select New Function from the Graph menu and create your own function (e.g.
f(x)=ex)
11. 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:
f  x  = ex
input x = 3.63
input
-10
x
-5
5
10
5
10
f input x = 37.85
output
-10
-5
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:
12. Select the function value that you calculated in step 11 and then choose Mark Scale
Factor from the Transform menu.
13. Double click on the origin point of your output axis. This selects it as a center of
dilation.
14. 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.
Olive & Oppong: Transforming Mathematics with GSP 4, page 85
15. 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.)
f  x  = ex
input
x
= 2.01
input
-10
f input
-5
x
=
x
5
10
7.47
output
-10
-5
5
f(x)
10
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.
Olive & Oppong: Transforming Mathematics with GSP 4, page 86
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
Olive & Oppong: Transforming Mathematics with GSP 4, page 87
in put
C
h
h(C)
ou tpu t
in put
i
i(D)
ou tpu t
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.
Note: It is very important to make sure that the unit scales on both dynagraphs are the
same and that the origins are lined up vertically, otherwise the output values of h(x) and the
input values of i(x) will not be the same!
The process you used to compose i(h(x)) can be used to compose more than two
functions. Working from the input point of the outer function, split it from its axis and merge
it with the output point of the next function, then split the input point of this next function
from its axis and merge that point with the output point of the next inner function, and so on.
Experiment by composing several of the functions on page 2 of Dynagraphs.gsp. Investigate
the ranges of the composed functions and compare them to the ranges of the individual
component functions.
Assignment 7.3
Create three functions of your own, each of which belongs to a different family (e.g,
step, quadratic, and trigonometric) and investigate the composition of your three
functions. Write-up your investigations, highlighting any interesting or surprising
characteristics you discovered for your particular composition.
Asymptotic Behavior with Dynagraphs
Recall that in the Cartesian (2-D) representation of the graph of a function, some
functions have what are called asymptotes. These are imaginary lines that the function tends
towards as the input approaches a discontinuity that produces an undefined output (e.g. the
function 1/x as x approaches zero). The following problems (designed by Dr. Tanya Cofer
Olive & Oppong: Transforming Mathematics with GSP 4, page 88
for students in a Mathematics Education course at the University of Georgia) ask you to
explore how asymptotic behavior appears within a dynagraph:
1. How would you characterize vertical asymptotes using the dynagraph representation?
(see function w on page. 2 of the Dynagraph.gsp sketch). Come up with other
rational functions that behave differently about the vertical asymptote.
2. How would you characterize horizontal asymptotes using this representation? Start
by analyzing the dynagraph of: y=(2x–3)/(x+1) (look at its “behavior at infinity”)
then come up with other examples of functions with horizontal asymptotes.
3. How would you characterize slant asymptotes using this representation? Start by
analyzing the dynagraph of: y=(x2+3x-5)/(x+1) (look at its “behavior at infinity”)
then come up with other examples of functions with slant asymptotes.
4. How would you characterize the asymptotes of: y=(x2+3x-5)/(x2+1)? What do you
notice about the range of this function?
From Dynagraphs to Cartesian Coordinates
The transition from the parallel number lines representation of functions to the more
traditional Cartesian Coordinate representation can be made dynamically using your GSP
dynagraph. Even though both GSP 3 and GSP 4 have built-in coordinate systems, it can be
enlightening and interesting to create your own two-dimensional coordinate system from
your one-dimensional number line or dynagraph. What you will be doing, in fact, is
transforming a mathematical mapping from R1 to R1 (the real numbers) into a mapping from
R1 to R2 (2-space).
A pair of parallel number lines can be transformed into non-parallel, intersecting
number lines to form coordinate axes in 2-space. The axes can be oblique as well as
rectangular, and do not have to share a common origin. One simple way to construct such a
flexible coordinate system is to construct a rotated image of the second (output) number line
about its origin. The following steps are provided as a guide:
1. Move the output axis of your dynagraph above the input axis.
2. Mark the origin of the output axis as a center of rotation.
3. Place a free point somewhere above the unit point of the output axis and label this
point tilt.
Olive & Oppong: Transforming Mathematics with GSP 4, page 89
4. Select (in this order) the unit point and origin point of your output axis AND the
point tilt and choose Mark Angle from the Transform menu.
5. Select the output axis and the output point (f(x)) and choose Rotate from the
Transform menu. The output axis and point should rotate about its origin by the
marked angle. Your dynagraph will look something like figure 7.13.
6. Label the rotated f(x) point and hide the original output axis and point.
f(x)
tilt
output
-10
-5
input
-10
-5
f(x)
0' 1'
0 1
inputx = 4.80
5
10
15
10
15
x
5
fx = 2
 x-32


f inputx = 6.46
Figure 7.13: A Rotated Dynagraph Output Axis
By moving the two origin points together you will have what is called an Oblique
Cartesian Coordinate System, named after the French philosopher and mathematician, René
Descartes (1596-1650) who is considered one of the fathers of analytic geometry. When the
two axes are perpendicular to one another, the system is called a Rectangular Cartesian
Coordinate System. Research the web to find out about René Descartes and his coordinate
system. Did Descartes ever use oblique coordinates? If so, for what purpose?
A point is plotted in these systems by constructing lines through the input and output
points that are parallel to the other axis. That is, a line through x parallel to the rotated output
axis and a line through f(x) parallel to the horizontal input axis. Where these two lines
intersect will be the point in the coordinate plane corresponding to the ordered pair (x, f(x)) or
(x, y) using the conventional label of y for the output axis. It is important to realize that the
intersection of these two lines is what coordinates the motion of the input and output points
Olive & Oppong: Transforming Mathematics with GSP 4, page 90
on their respective axes. Thus, the coordinate graph of a function is the locus of this
coordinate point; i.e. the path of the coordinate point as the input point x varies.
Hide the horizontal output axis and move the two origin points together. [NOTE: You
can use a Movement button to move 0’ to 0 automatically and accurately. Select 0’and 0 IN
THAT ORDER and choose Edit/Action Buttons/Movement from the menu bar. Click OK
to create your movement button. Click on the movement button to move 0’ to 0.] Construct
the parallel lines on your oblique coordinate system and the coordinate point (x,y) at their
intersection. Trace this point as you move your input point x. You can also construct the
locus of this coordinate point by selecting point x and point (x,y) and then choosing Locus
under the Construct menu. You should have something like the picture in figure 7.14.
Move 0' -> 0
(x,y)
f(x)
tilt
input
-10
-5
x
0 1
0'
inputx = 4.80
5
fx = 2
10
 x-32

15

f inputx = 6.46
Figure 7.14: Oblique Coordinate Graph of a Quadratic Function
Investigate the properties of this oblique graph of a quadratic function.
 Where does it meet or intersect the x-axis?
 Where does the graph intersect the y-axis?
 Do these points change when you change the tilt angle of the y-axis?
 Does this graph pass the “vertical line test” for a function?
 What would be an appropriate “test” for a function graphed on oblique axes?
Olive & Oppong: Transforming Mathematics with GSP 4, page 91
Edit your function by double clicking on the f(x) expression and test your conjectures with
different kinds of functions.
Sketchpad has a built-in rectangular coordinate graphing system that automatically
constructs the locus of the ordered pair (x, f(x)) when a function is plotted using the Plot
Function option under the Graph menu. Select the function expression in your sketch and
plot its graph using this feature of Sketchpad. You should have a figure that looks something
like figure 7.15.
Move 0' -> 0
(x,y)
f(x)
tilt
input
-10
-5
x
0 1
0'
inputx = 4.76
5
fx = 2 x-3 2
10

15

f inputx = 6.21
Figure 7.15: Oblique and Rectangular Graphs of a Quadratic Function
Rotate the tilt point until both graphs coincide. Move your input point x and observe
how the plotted point (x,y) moves along the plotted function graph. For the next part of this
chapter will shall use Sketchpad’s function plotting feature to investigate transformations of
the quadratic function.
Activity 7.4: Transformations of the Quadratic Function
In a new sketch, plot the function f(x)=x2 using Plot Function under the Graph menu.
A parabola should have been plotted. Place a free point on this parabola using the Point tool
(just click on the parabola with the Point tool). Label this point A. Measure the coordinates
of point A (select point A and then select Coordinates from the Measure menu). Observe
how the coordinates change as you move point A along the parabola. We shall now use this
point on the graph of f(x) to translate the parabola (function plots themselves cannot be
transformed directly).
Olive & Oppong: Transforming Mathematics with GSP 4, page 92
In order to translate the parabola, we need to define a vector of translation. We shall then
use this vector to translate point A on the parabola

Create a free point somewhere in your sketch and label it V.

Select the origin point and point V IN THAT ORDER and choose Mark Vector from
the Transform menu.

Select point A and choose Translate under the Transform menu. Display the label of
the translated point A’.

Select A’ and choose Trace Point under the Display menu.
Move point A. What shape does the translated point, A’ trace out? What do you notice about
your free point V relative to the trace of A’?
Measure the coordinates of points V and A’. Move point A. What do you notice about the
coordinates of A, V and A’? Move point V. What do you notice about the coordinates of A, V
and A’? You may want to measure the abscissa and ordinates of points A and V and do some
calculations with these to check your conjecture concerning relations among these
coordinates.
In order to more closely observe and keep track of the changes in the parabolic graph
of the quadratic as point V is moved we can construct the locus of point A’ (instead of tracing
its path) as A is moved. First choose Erase Traces under the Display menu, then select A’
and turn off its trace by choosing the checked Trace Point under the Display menu. Select
points A’ and A then choose Locus from the Construct menu. Your sketch should look
something like figure 7.16. Move point V around your sketch. What do you notice about
point V and the translated parabola?
Olive & Oppong: Transforming Mathematics with GSP 4, page 93
f x = x2
6
A: (1.44, 2.08)
V: (2.26, -2.89)
4
xA = 1.44
xV = 2.26
2
yA = 2.08
A
yV = -2.89
-10
-5
xA+xV = 3.70
5
A': (3.70, -0.81)
10
A'
yA+yV = -0.81
-2
V
-4
Figure 7.16: Translated Quadratic Function
-6
Assignment 7.4
Challenge: You probably realized that point V remained the vertex of the translated
parabola no matter where you moved it. Use the abscissa and ordinate measures of V to form
a new function whose plot will always be coincident with the locus of A.’ Call your new
function g(x). [Hint: Recall the vertex form of a quadratic function.]
[Note: To use any measure or parameter in the definition of a new function, just click on the
measure or parameter when the New Function calculator window is open. You can also
click on an existing function expression to use that in the definition of a new function.]
The construction above created a translation of a quadratic function and its parabolic
graph. We shall now use a parameter to create a dilation of the quadratic function and its
parabolic graph.

Choose New Parameter from the Graph menu and change its name to “d” and its
value to 2.

Select this new parameter and choose Mark Scale Factor under the Transform menu.

Double click on the origin point of your axes to mark it as a center for dilation.

Select point A and choose Dilate under the Transform menu. Click the Dilate button
to dilate A by the marked ratio, scale factor d. A new point should have appeared,
twice as far from the origin as A.

Change the label of this new point to Ad by typing “A[d]” in the label dialog window.
Also change the label of your translated point, A’ to At the same way.
Olive & Oppong: Transforming Mathematics with GSP 4, page 94

Select Ad and A, then choose Locus under the Construct menu to construct the locus
of your dilated point as A moves on the original parabola.
Your sketch should look like figure 7.17 at this point. Select the parameter d, and then
change its value by using the + and – keys on your keyboard. What happens to the dilated
parabola? [Note: The default step for changing a parameter with the +/- keys is one unit.
You can change this step. Select parameter d and choose Properties from the Edit menu.
Select the Parameter tab on the pop-up window. At the bottom of this dialog box change
the “Change by:” value to 0.1 units for Keyboard (+/-) adjustments. Click on the OK
button.] Now press the + key several times. What do you notice?
12
10
Ad
8
f x =
x2
A: (2.07, 4.27)
6
V: (3.14, -2.54)
A
4
xA = 2.07
d = 2.00
xV = 3.14
yA = 4.27
2
At
yV = -2.54
-10
0
-5
xA+xV = 5.21
yA+yV = 1.73
At: (5.21, 1.73)
5
10
15
-2
V
-4
-6
Figure 7.17: Dilated and Translated Quadratic Functions
Assignment 7.5
-8
Challenges:
1. Create a new function using parameter d whose graph will coincide with the dilated
-10
parabola of f(x). Call your new function h(x).
2. Create a new quadratic function based on the free point V as the vertex of your
parabola and the parameter d as the dilation factor, that will be the translation of
your function h(x) by the vector 0V. Call your new function j(x).
3. How does j(x) compare to the general quadratic function in vertex form: y=a(xh)2+k, where (h, k) are the coordinates of the vertex of the parabola? What is the
relationship between the parameter a in this general vertex form and your parameter
d?
Olive & Oppong: Transforming Mathematics with GSP 4, page 95
Extra Challenge: Starting with a free point V as the vertex of your parabola, create a
general quadratic function that can be moved around the screen simply by moving it’s
vertex, V, and can change its “openness” from very narrow opening up to very wide
and eventually to opening down by moving a control point on the parabola. This
control point should move with the parabola when you move point V (see figures 7.18a
and 7.18b for two possible states of this completely dynamic quadratic).
12
12
h = 1.73
k = 2.54
f x = x-h 2 +k
h = 1.73
k = 10.44
f x = x-h 2 +k
10
y A = 3.46
y A = 10.02
8
y A-k = 0.92
V
10
A
8
y A-k = -0.42
6
a = 0.92
a = -0.42
gx = a x-h 2 +k
6
gx = a x-h 2 +k
A
4
4
V
2
2
-5
5
-5
5
Figure 7.18a and 7.18b: A dynamic quadratic graph controlled by points V and A
Need to add: From vertex-form of quadratic function to Completing the Square to find roots
and eventually the quadratic formula
Draft: for f(x)=a(x-h)2 + k -- where (h, k) are the coordinates of the vertex of the parabola,
We can solve for roots by putting f(x)=0: a(x-h)2 + k=0  a(x-h)2=-k  (x-h)2=
x-h= 
k
a
, thus x=h 
form.
k
a
k
a

. This can be thought of as the quadratic formula in vertex

Converting from vertex form to standard form: f(x)=ax2+bx+c by expanding the binomial


b2
b
a(x-h)2 and equating like terms, gives h= , k=c. Substituting for h and k in the
4a
2a
equation x=h 
k
a
in terms of a, b and c leads to the standard quadratic formula for the
b 2  4ac
 b  
roots of a quadratic function: x =
.
2a

Completing the Square when starting with the standard form is really converting the standard
form into vertex form, and thus (again) finding h and k in terms of a, b and c.

Olive & Oppong: Transforming Mathematics with GSP 4, page 96
Reflections
Dynagraphs were very probably a new way of representing and playing with
functions for you. In what ways did they enhance your own concepts and ideas about
functions? Would you use these dynamic representations with your students? Why or why
not?
References
Goldenberg, E. P., Cuoco, A. A., and Mark, J. (1992). Making Connections with Geometry.
A plenary paper presented to the Geometry Working Group at the Seventh International
Congress on Mathematics Education (ICME-7), Quebec, Canada.
Lin, P-P. and Hsieh, C-J. (1993). Parameter Effects and Solving Linear Equations in
Dynamic, Linked, Multiple Representation Environments. The Mathematics Educator,
4, 1, 25-33.
Olive, J. (1993). Technology and School Mathematics. The International Journal of
Educational Research , 17, 5, 503-516.