Transformations: stretches

Pulling the graph like dough

A stretch pulls the graph away from one axis, or squashes it towards it.

Same rule as for translations:

outside the bracket acts on $y$ , and does what it says;
inside the bracket acts on $x$ , and does the opposite.

Try it with numbers first. Say the point $(4, 5)$ is on $y = f(x)$ , so $f(4) = 5$ .

On $y = 3f(x)$ : at $x = 4$ , the height is $3 \times f(4) = 15$ . So $(4, 5)$ moves to $(4, 15)$ . Heights are tripled.
On $y = f(2x)$ : which $x$ gives $f(2x) = f(4)$ ? We need $2x = 4$ , so $x = 2$ . So $(4, 5)$ moves to $(2, 5)$ . The $x$ -value is halved, not doubled.

In general, with $a$ standing for a positive number ( $a > 0$ ; negative values are covered below):

Equation	Transformation	Point $(p, q)$ goes to
$y = af(x)$	stretch parallel to the $y$ -axis, scale factor $a$	$(p,\ aq)$
$y = f(ax)$	stretch parallel to the $x$ -axis, scale factor $\dfrac{1}{a}$	$\left(\dfrac{p}{a},\ q\right)$

The scale factor is what every distance from the fixed axis is multiplied by. A factor of 3 makes those distances 3 times as big; a factor of $\tfrac12$ halves them.

So $y = 3f(x)$ triples every height, while $y = f(3x)$ squeezes the graph to a third of its width.