Are transformations of functions giving your students trouble? Try a covariation approach.

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, 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/ath 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?

A covariation approach helps students to focus on mathematical relationships. Using relationships is more powerful than trying to remember rules.