# Make Algebra & PreCalculus Come Alive Using Cabri Geometry

# Make Algebra & PreCalculus Come Alive Using Cabri Geometry

Function Families & the Crosshair Technique

As handheld graphing technology has become more prevalent, our students are not
spending

significant time graphing functions by pencil and paper techniques.
Nevertheless, we still want

them to recognize the graphs of key function families, and understand how
changing the constants

in the equation will affect the graph of the function.

On a TI-82/83/83plus, the students can modify various parameters in the Y=
window to explore

how these adjustments affect the graph. But this requires the student to switch
back and forth

between the Y= and graphing windows. Figures and graphs constructed in the Cabri
Geometry II

program, however, can dynamically respond to changes students make in equation
parameters. I

have been using the power of Cabri on a computer to demonstrate graphs of many
function

families to my students. An additional advantage of using the computer includes
its superi

or display qualities, with color available and a larger screen area.

The basis for these demonstrations is the “Crosshair” technique taught in the
Connecting Algebra

& Geometry summer Institute (CAG) developed by Teachers Teaching with
Technology.

This technique turns the Cabri Geometry screen into a function grapher, by using
Cabri’s

embedded coordinate geometry environment. Once this technique is mastered, it
can be used to

create graphs of any function.

Using the “Crosshair” technique to graph any function and explore its
properties:

1. On a fresh screen, Show Axes and Define Grid by clicking on the axes.

2. Use the Segment tool to lay a segment on the X-axis. This will put a boundary
on your x-

value, the dependent variable for the function .

3. Put a Point on the segment, not on the X-axis. Find the Coordinates of this
point.

4. Use Numerical Edit to place as many editable numbers on the screen as needed
in the

equation. For example, the quadratic function f(x) = ax^{2 }+ bx + c requires 3
numbers, for a,

b, and c. Use numbers with one decimal place , like 1.1, to easily manipulate
later. (see fig.1)

Figure 1

5. Use a Comment box to write the general equation. When
one of the previously created

numbers is needed in the equation, click on it to
include it. Then delete the original numbers.

The new numbers in the equation
are now editable.

6. Use the Calculate tool to evaluate the y-value according to the equation. To
do this, click on

the numbers in the equation and the point’s x-value as needed.
Click in the result box and

drag the result onto the screen. (see fig. 2)

Figure 2

7. Use Measurement Transfer: click on the result of the calculation, then on the
Y-axis. A new

point appears at that y-value.

8. Create the crosshairs: construct Perpendicular Lines through the x and y
points perpendicular

to the appropriate axis. Then place a Point at the
intersection of these. (see fig. 3)

Figure 3

9. Find the Locus of this point with respect to the point
on the X-axis (see fig. 4). Do this by

clicking first on the intersection point
(the object for which to construct the locus) then on the

X-axis point (the
object that moves with respect to some path, in this case the original

segment
placed on the X-axis).

10. To edit a number in the equation, double-click on it
and use the scroll arrows that appear.

Explore what happens to the graph as the
numbers change. Since the graph updates

automatically, students see the graph
“come alive.”

Positive “a” value (Fig. 4)

Figure 4

Negative “a” value (Fig. 5)

Figure 5

Changing the value of “c” (Fig. 6)

Figure 6

My personal favorite is the basic power function,
. By
letting n equal any integer or a

fraction between one and zero, students can see
patterns in graphs of power functions, inverse

variation, and radical functions .

As I adjust the exponent, I ask the students to predict the effect on the graph.
I also ask them to

explain what happens to the graph when the exponent equals
one or zero. The interactive

computer display stimulates a fruitful class
discussion about the properties of the graphs of power

functions, the patterns exhibited as the exponent changes, and the connection
between

and
fractional exponents. It is a great opportunity to bring together several topics
from Advanced

Algebra or Pre -Calculus in one animated lesson. (see figs. 7-10)

Figure 7: positive even integer exponent | Figure 8: positive odd integer exponent |

Figure 9: inverse variation | Figure 10: square root function |

The Crosshair technique works for any type of function.
Try to build linear , polynomial,

trigonometric , exponential or logarithmic
functions using this method. It can even be used to

build conic sections in a
piecewise manner. These lessons are successful as classroom

demonstrations in
which students take notes on the graphs and discussion, or as lab activities for

individuals or pairs of students. I create the Cabri file in advance, so class
time is focused on the

dynamic view of function behavior.

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