## Consider this problem:

Sketch a graph of a function *y=f(x)*.

Now sketch a graph of *y=f(***a**x) 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(***a**x) 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(***a**x) represents a relationship between two “*things*” that can change: “*y*” and “*ax*“

The notation *y=f(***a**x) 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(***a**x), 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?

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