Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.
Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!
GeoGebraBook: Exploring Linear Functions,
Interactive Linear Function Graph: Slope-Intercept Form
Interactive Linear Function Graph: Point-Slope Form
Continue reading Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube
Check out this GeoGebraBook of nine applets that will help you explore Unit Circle Symmetries. It contains three applets per type of symmetry on the unit circle, one focusing only on the unit circle, and the other two linking unit circle properties to patterns in the graphs of the sine and cosine functions.
When two angle expressions, such as and , exhibit symmetry on the unit circle, an understanding of unit circle symmetries and reference angles often allow function arguments to be simplified. Mastery of symmetries and reference angles will also be very handy when expanding inverse trigonometric function results to describe all possible answers to a problem.
Suggestions for improvements to these applets, or additional applets, are always welcome via comments on this post.
Suppose nobody had ever thought of measuring the size of an angle, and someone asked you “How can I describe the size of an angle?” What approach might you take in answering this question?
You might start by arbitrarily picking some angle, any angle, such as angle ABC in the image below, and call its measure “1”. All other angles could be
Continue reading Angle Measures
For the approach I now prefer to this topic, using transformation equations, please follow this link: Function Transformations: Translation
A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. A function can be translated either vertically, horizontally, or both. Other possible “transformations” of a function include dilation, reflection, and rotation.
Imagine a graph drawn on tracing paper or a transparency, then placed over a separate set of axes. If you move the graph left or right in the direction of the horizontal axis, without rotating it, you are “translating” the graph horizontally. Move the graph straight up or down in the direction of the vertical axis, and you are translating the graph vertically.
In the text that follows, we will explore how we know that the graph of a function like
which is the blue curve on the graph above, can be described as a translation of the graph of the green curve above:
Describing as a translation of a simpler-looking (and more familiar) function like makes it easier to understand and predict its behavior, and can make it easier to describe the behavior of complex-looking functions. Before you dive into the explanations below, you may wish to play around a bit with the green sliders for “h” and “k” in this Geogebra Applet to get a feel for what horizontal and vertical translations look like as they take place (the “a” slider dilates the function, as discussed in my Function Dilations post).
Continue reading Function Translations: How to recognize and analyze them
The greatest value of GeoGebra, or Geometer’s Sketchpad, or other such packages, in my eyes is their ability to help students dynamically visualize the effect each constant has on the graph of an equation. I find them an invaluable “thinking aid” as I ponder a new equation form, and they help me formulate my own answers to questions such as “why does it do that?”
Check out applets that help students explore the relationship between function parameters and their graphs:
GeoGebraBook: Exploring Linear Functions, which contains
GeoGebraBook of Quadratic Applets, which contains
Exponential and Logarithmic functions:
GeoGebraBook of Exponential and Logarithmic Applets, which contains
GeoGebraBook for Exploring Trig Functions, which contains
Unit Circle Symmetries:
GeoGebraBook of Unit Circle Symmetry applets, which contains: