## 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(*compare to your graph of

**a**x)*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(***a**x) represents a relationship between two “*things*” that can change: “*y*” and “*ax*“

**a**x)

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.”