Unit 3

Transformations

By examining the graph of a function and its image, the equation of the transformed function can be determined.

f (x) = x^{2}

has been transformed to

g (x)

. Write a possible equation for

g (x)

Two solutions are possible.

Case 1: A vertical stretch was used.

Locate key points on

f (x)

and the corresponding points on

g (x)

(points that have the same

x

-coordinates). Compare the

y

-coordinates.

$x$	$f (x) = y$	$g (x) = y$	Description
$x = 0$	$f (0) = 0$	$g (0) = 0$	This is an invariant point because it did not change under the transformation.
$x = 1$	$f (1) = 1$	$g (1) = 4$	The $y$ -value of $f (x)$ has been multiplied by $4$ to get the $y$ -value of $g (x)$ .
$x = 2$	$f (2) = 4$	$g (2) = 16$	The $y$ -value of $f (x)$ has been multiplied by $4$ to get the $y$ -value of $g (x)$ .
$x = - 2$	$f (- 2) = 4$	$g (- 2) = 16$	The $y$ -value of $f (x)$ has been multiplied by $4$ to get the $y$ -value of $g (x)$ .

The table indicates that the transformation is a vertical stretch about the

x

-axis by a factor of

4

. The equation of the transformed function is

g (x) = 4 x^{2}

.

Case 2: A horizontal stretch was used.

Locate key points on each graph that have the same

y

-coordinates. Compare the

x

-coordinates.

$y$	Points on $f (x)$	Corresponding points on $g (x)$	Description
$y = 0$	$(0, 0)$	$(0, 0)$	This is an invariant point because it did not change under the transformation.
$y = 1$	$(- 1, 1)$ and $(1, 1)$	$(- \frac{1}{2}, 1)$ and $(\frac{1}{2}, 1)$	The $x$ -values of $f (x)$ have been multiplied by $\frac{1}{2}$ to get the $x$ -values of $g (x)$ .
$y = 4$	$(- 2, 4)$ and $(2, 4)$	$(- 1, 4)$ and $(1, 4)$	The $x$ -values of $f (x)$ have been multiplied by $\frac{1}{2}$ to get the $x$ -values of $g (x)$ .
$y = 16$	$(- 4, 16)$ and $(4, 16)$	$(- 2, 16)$ and $(2, 16)$	The $x$ -values of $f (x)$ have been multiplied by $\frac{1}{2}$ to get the $x$ -values of $g (x)$ .

The transformed function,

g (x)

, can also be interpreted as

f (x)

with a horizontal stretch of

\frac{1}{2}

applied.

The equation showing a horizontal stretch about the

y

-axis by a factor of

\frac{1}{2}

g (x) = {(2 x)}^{2}

.

Note that for

f (x) = x^{2}

, a vertical stretch by a factor of

4

is the same as a horizontal stretch by a factor of

\frac{1}{2}

.

Vertical stretch about the

x

-axis by a factor of

4

g (x) = 4 x^{2}

.

Horizontal stretch about the

y

-axis by a factor of

\frac{1}{2}

g (x) = {(2 x)}^{2} = 4 x^{2}