Transformations of graphs

The four graph transformations and the two reflections — why a change "inside" the function behaves backwards — and how to combine them.

Pure · Algebra & functions

Transformations of graphs

The four graph transformations and the two reflections — why a change "inside" the function behaves backwards — and how to combine them.

Why it works

The whole topic turns on one idea: a change outside $f$ acts on $y$ and behaves as you'd expect; a change inside the bracket acts on $x$ and behaves backwards.

New equation	Effect	Direction
$y = f(x) + a$	translation, $a$ up	outside — as expected
$y = a\,f(x)$	vertical stretch, scale factor $a$	outside — as expected
$y = f(x + a)$	translation, $a$ to the left	inside — backwards
$y = f(ax)$	horizontal stretch, scale factor $\tfrac1a$	inside — backwards
$y = -f(x)$	reflection in the $x$ -axis	outside
$y = f(-x)$	reflection in the $y$ -axis	inside

Why is inside backwards? Take

y = f(x + 2)

. To read off the value the original curve had at

x = 0

, you now only need

x = -2

(so the bracket reads

0

) — every feature arrives

2

earlier, i.e. shifted left. Likewise

f(2x)

reaches at

x = 1

what

f

reached at

x = 2

: everything is squashed to half the width — scale factor

\tfrac12

, not

2

.

Working with a single point. A transformation moves every point, so you can track just the one you care about (a maximum, an intercept). Apply the outside change to its

y

-coordinate and the inside change to its

x

-coordinate.

Combining transformations. For something like

y = -f(x - 3) + 1

, deal with each piece:

x - 3

(inside) → right

3

; the leading

-

(outside) → reflect in the

x

-axis;

+1

(outside) → up

1

. Keep the inside and outside effects separate and you won't mix up which coordinate each one changes.