Consider this problem:
Sketch a graph of a function y=f(x).
Now sketch a graph of y=f(ax) for some constant value a>1.
Reflect.
What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? Did you choose a particular value for a? How does your graph of y=f(ax) compare to your graph of y=f(x)? What is similar? What is different?
It is no secret that students have difficulty making sense of transformations of functions. A covariation approach can help. Here’s how.
A graph of y=f(x) represents a relationship between two “things” that can change: “y” and “x.”
Think about a graph of y=f(x) as representing a relationship between two “things” that can change: “y” and “x“
The notation y=f(x) means that y is a function of x. Functions are special kinds of relationships between two “things” that can change. When y is a function of x, as the values of x change, the values of y change in predictable ways.
A graph of y=f(ax) represents a relationship between two “things” that can change: “y” and “ax“
The notation y=f(ax) means that y is a function of ax. When y is a function of ax, as the values of ax change, the values of y change in predictable ways. Because a is some constant value greater than one, a does not change. Therefore, the changing values of ax depend on changes in the values of x.
Working from a relationship between “y” and “x,” students can make sense of a relationship between”y” and “ax.”
For a function y=f(x), as the values of x change, the values of y change along with them.
For a function y=f(ax), the values of x change by a factor of “a.” This means that “1/a“th of the change in x will yield the same amount of change in y as for y=f(x).
Using relationships between changing “things” can help students to make sense of graphs of transformations of functions.
For example, sketch a graph y=f(x).
Now sketch a new graph with this constraint: “1/2 the change in x will produce the same change in y as for the original function.”
Reflect.
What kind of graph did you sketch for y=f(x)? How did you decide what to sketch? How does your new graph compare to your graph of y=f(x)? What is similar? What is different?
So what differences have you seen in learner thinking with this approach?
With a covariation approach, I have seen students focus more on a relationship that results in a particular graph shape. For example, students say things such as “distance keeps on going, and the height goes up and down.”