Function Transformations

Explain why

shifts the graph right by 3, rather than left.

If

is on $y=f(x)$, then for the transformed graph we need

so

Thus

Therefore the graph shifts right by 3.

If

is on the original graph, then

gives

Thus

So all $x$-coordinates are divided by 2.

Describe the transformation from

to

Describe

The point

lies on

Find the corresponding point on

Starting with

describe

If

lies on $y=f(x)$, find the corresponding point on

Reflecting the graph of $y=f(x)$ in both coordinate axes gives

This is equivalent to a reflection about the origin, or a $180^\circ$ rotation about the origin.

If

then

produces exactly the same graph.

So reflection in the $y$-axis leaves an even function unchanged.

If

then

Thus reflecting an odd function in both axes leaves its graph unchanged.

If $f$ has domain $D$, then

requires

Similarly, if the range of $f$ is $R$, then the range of

is obtained by applying

to $R$.

For complicated expressions such as

it is often safer to take

on the original graph, solve

and transform $v$, rather than trying to perform several transformations mentally.

Solving,

so

For the $y$-coordinate,

Therefore,

Incorrect:

Correct:

Incorrect:

is a horizontal stretch by 2.

Correct:

The $x$-coordinates are divided by 2.

Do not interpret it as simply:

horizontal compression by 2, then right 6.

Factor first:

The shift parameter is 3, not 6.