LOCUS AND GEOGEBRA
Garrett Delk
gdelk71687@csu.fullerton.edu
Armando M. Martinez-Cruz
amartinez-cruz@fullerton.edu
Department of Mathematics
Cal State Fullerton
Presented at
GSDMC
34th Annual Mathematics Conference
San Diego, CA
Saturday Feb. 2nd, 2013
Legend for Geogebra Tools
Move- Drag or select objects (ESC)
Move Graphics View – Moves the whole graph/view
New Point – Creates a new point by clicking on line, function or curve
Perpendicular Line – Creates a perpendicular line
Locus – Creates a locus after selecting locus point then object
Tangents - Creates a tangent line to a specific object given a point.
Circle with Center through Point – Creates a circle when given a center
Circular Arc with Center between two Points – Creates an
arc when given a center and two arbitrary points.
Angle – Creates an angle
Slope – Gives slope of a line after selecting a line
Using Locus to Construct the Sine Function
Step 1: Open Geogebra and shorten the algebra window by dragging the divider.
Step 2: Select the “New Point” button and construct point A at the origin.
Step 3: Select the “Circle with Center through Point” button and construct a circle
with a radius of 3. To do this, first select the center of your circle (point A),
then drag the cursor to create a circle with radius 3. Note that this will create
another point B once the circle is constructed.
Notice that the algebra window on the left will give you the equation of your circle
labeled as 𝒄: 𝒙𝟐 + 𝒚𝟐 = 𝟗 .
Step 4: Select the “New Point” button and place an arbitrary point on the
circumference of the circle. Notice that the software labels it as C.
Step 5: Select the “Circular Arc with Center between two Points” button under the
“Circle with Center through Point” button. To construct the arc, first select
the center of circle (point A), then select point B and point C (in that order).
This process has constructed arc BC on top of the circle.
Notice that the algebra window has a new parameter labeled d. This parameter
gives the length of arc BC in radians. We will use d to construct a point on the locus
called LOCUS POINT.
Step 6: In the input bar, enter the point (d, y(C)). This will create a point D on the
plane. Make sure that the coordinates are case sensitive.
In step 6 we created a point with x-coordinate given by the length of the arc, namely
d. The y-coordinate is given by the y-coordinate of the point C. Drag the point C
around the circle using the pointer button and notice that the point D moves
accordingly.
Step 7: Select the “Locus” button under the “Perpendicular Line” button. Next
select D (locus point), then select point C.
VOILA! We have constructed the Sine function using the locus tool on Geogebra. If
we drag point C we can see point D moving along the Sine curve.
EDIT: To construct the Cosine and Tangent functions, follow the same steps except
in step 6 change the input to (d, x(C)) and (d, (y(C)/x(C)) respectively.
Using Locus to Construct the Derivative of a Function
We now construct the derivative of an arbitrary polynomial using the geometric
locus and Geogebra tools.
Step 1: Open Geogebra and shorten the algebra window by dragging the divider.
Step 2: In the input bar enter an arbitrary third degree polynomial, or enter the
following function: 𝒇(𝒙) = 𝟑𝒙𝟑 − 𝟗𝒙𝟐 + 𝟑𝒙 − 𝟕.
Step 3: Construct point A anywhere on the graph of the polynomial.
Step 4: Construct a tangent line to the function using the “Tangents” button under
the “Perpendicular Line” button. To do this, select point A then the graph of
the polynomial.
Step 5: Measure the slope of the tangent line using the “Slope” button under the
“Angle” button. To do this, just select the tangent line.
Notice in the algebra window we have: the coordinates of point A, the equation of
the tangent line (line a) as well as its slope (denoted by m). These are dependent
objects that change when point A is moved along the graph of the function. We will
use the slope of the tangent line to create a new point on the locus called the LOCUS
POINT.
Step 6: In the input bar, enter the point (x(A),m). This will create a point B on the
plane. Make sure that the coordinates are case sensitive. If you cannot see
point B, drag point A.
Here, we created a point with x-coordinate given by the x-coordinate of A. The ycoordinate is given by the slope of the tangent line, namely m. Drag the point A
along the function using the pointer button and notice that the point B moves
accordingly. For instance, note that if the slope is positive, then B is in the I or II
quadrants. When the slope is negative, B is in the III or IV quadrants.
Step 7: Construct the locus by using the “Locus” button under the “Perpendicular
Line” button. To do this, select point B (locus point), then select point A.
VOILA! We have constructed the derivative of our function using the locus tool on
Geogebra. If we drag point A we can see point B describing the graph of the
derivative of f(x).
Using Locus to Construct the Parabola
The parabola is defined as the locus of points that are equidistant to a fixed point
(called the Focus) and to a fixed line (called the Directrix).
Steps
1.
2.
3.
4.
5.
6.
7.
Construct a point and label it as Focus.
Construct a line AB and label it as Directrix.
Construct another point on the Directrix and call it C.
Construct a perpendicular line to the Directrix through C.
Construct the segment Focus-C.
Construct the perpendicular bisector of segment Focus-C.
This perpendicular bisector intersects the perpendicular to the Directrix at a
point, say P.
8. Construct the locus of P when C moves along the Directrix.