In fact, you can see for yourself: in the applet below, drag the axes' unit (the "1" on the x-axis) left or right, and watch the `a` in the equation change while the parabola's shape remains absolutely constant. Select either one and click on or near the origin, and see how the same parabola's shape appears to change.Äownload the file: GeoGebra | Cabri Conversely, look at how different equations can yield the same shape: In the applet below, click on the magnifying glass icon to switch between the Zoom In and Zoom Out tools. The same approach works on a document camera: place the original next to the projected image of a photocopy of itself.) Compare the two transparencies, which seem to have very different shapes, but clearly must represent the same equation. enter as is: Crossu,v This is the cross product of two vectors Step 5. u,v This is the dot product of two vectors Step 4. Plotting Vectors in 3 Dimension Preparation: Delete all existing vectors.
#Geogebra classic drag axes software#
Use another transparency to trace a piece of the projection, like the one below. As you can see using Geogebra software tool, it is easy now to create 3D. (Dan Bennett suggests a dramatic illustration of this: make a transparency of a figure like the one above. Visual Arguments: Same equation, apparently different shapes: The reflection property suggests this construction of sorts: Therefore P' is P, and MP is tangent.Įxercise: Prove the reflection property of the parabola, assuming that the angles of incidence and reflection are determined with respect to the tangent to the parabola at the point of incidence. But the only way for P'T to equal P'T' is for T' to be T. Because it is on the parabola, it is equidistant from F and d, so if T' is the foot of a perpendicular from P' to d, we have P'T = P'F = P'T'. Because P' is on MP, it is equidistant from F and T. Indeed, assume MP intersects the parabola at P and another point P'.
![geogebra classic drag axes geogebra classic drag axes](https://www.geogebra.org/resource/atfumbnu/VJSMWHF5fgah1m5p/material-atfumbnu-thumb.png)
All but five students made classic mistakes, such as picturing the graphs as.
![geogebra classic drag axes geogebra classic drag axes](https://www.geogebra.org/resource/mq77gwax/QeB8aA7bTCP3kEOL/material-mq77gwax-thumb.png)
The key to the proof is realizing that MP must be tangent to the parabola. when teaching a unit on linear functions with GeoGebra, a dynamic algebra. That this works is readily proved using the above construction, if you assume a basic fact from optics: the angle of incidence equals the angle of reflection.
![geogebra classic drag axes geogebra classic drag axes](https://www.tentotwelvemath.com/wp-content/uploads/2020/11/IMG_20201106_120228932-1024x768.jpg)
Likewise, a light ray coming in parallel to the axis of symmetry will be reflected to hit the focus. A light ray originating at the focus will be reflected on the parabola and continue in a direction parallel to the axis of symmetry.