Implications for Instruction
Discovering what happens to graphs when constraints are placed upon them is a great way to talk about transformations in the mathematics classroom. If one understands transformations then they are able to predict what a graph may look like if they know the original or basic graph of the function. They may also learn to predict what the function is if they are given the graph.
Asymptotes are a great way to emphasize the domain and range of a function. Students should recognize that there are certain rules of mathematics that must always hold. One is that you can never divide by zero. Whatever value makes the function y = 1/x have zero in the denominator creates an asymptote at that point. Depending on if the asymptote is a vertical or if it is a horizontal one, it will tell the students about the domain and the range of the graph. While tutoring, this always seemed to be a difficult subject to teach because students would get confused by simply looking at the graphs. They would also get confused if domain meant the y values or the x values.
When discussing the sine function, the audience would probably be students learning trigonometry. The sine function does not have an asymptote but it does contain a limit. These are similar in the fact that there are values that the equation cannot give you. If a student understands this, then they are able to realize if values make sense.
Comparing and contrasting graphs will allow students to understand that the
transformations and properties of transformations hold for each type of function.
I probably would not have students do a trigonometric graph and a non trigonometric
graph to compare and contrast. I would probably start off with y = x^2 and y
= x^3. I would also give them a graph that has the transformations already done
to the graph and then ask them to tell me what the equation of the graph is.
If they understand the properties of transformations then they should be able
to do this activity.