Transformations: reflections

By the end you can sketch and describe $y = -f(x)$ and $y = f(-x)$ as reflections, find their equations, and say exactly which mirror line is involved.

Same rule as for translations:

a change outside the bracket, like $-f(x)$ , acts on the output $y$ ;
a change inside the bracket, like $f(-x)$ , acts on the input $x$ .

Here the change is a minus sign. A minus sign flips things, so both are reflections (mirror images):

Equation	What changes	Transformation	Point $(p, q)$ goes to
$y = -f(x)$	every height changes sign	reflection in the $x$ -axis	$(p,\ -q)$
$y = f(-x)$	every input changes sign	reflection in the $y$ -axis	$(-p,\ q)$

Points on the mirror line stay put:

under $y = -f(x)$ , the $x$ -intercepts don't move (their height is 0, and $-0 = 0$ );
under $y = f(-x)$ , the $y$ -intercept doesn't move (its $x$ is 0, and $-0 = 0$ ).

Put $x = -3$ into $y = f(-x)$ . You get $f(3)$ : the height $f$ had at $x = 3$ .

So whatever happened on the right now happens on the left, at the same height. That's a mirror standing on the $y$ -axis.

$y = -f(x)$ is different. It keeps every input where it was, but turns each height upside down. That's a mirror lying along the $x$ -axis.

Which transformation maps the graph of $y = f(x)$ onto the graph of $y = f(-x)$ ?

The point $(3, -4)$ lies on the graph of $y = f(x)$ . Find the corresponding point on the graph of $y = -f(x)$ .

Two mirrors