Transforming graphs

Once you know a parent graph, like $y = x^{2}$ , you know a whole family. Each change to the formula moves or stretches it in a predictable way.

y = x^{2}

in orange.

y = (x - 2)^{2} + 1

moves right 2 and up 1.

Change	Effect
$f (x) + k$	Up $k$
$f (x - h)$	Right $h$
$- f (x)$	Flip over the $x$ -axis
$a f (x)$ , $a > 1$	Taller, stretched vertically

Why does $x - 2$ go right? $(x - 2)^{2}$ reaches 0 when $x = 2$ . The output you used to get at 0 now comes 2 later.

Work through it

Inside: $x + 3$ is $x - (- 3)$ .left 3

Negative.flip downward

Outside.up 4

Vertex. $(- 3, 4)$

Inside first. $(x - 1)^{2}$

Stretch outside. $2 (x - 1)^{2}$

Move down. $y = 2 (x - 1)^{2} - 5$

Vertex. $(1, - 5)$

Thinking $(x - 2)^{2}$ moves left.

Inside changes behave backward: minus moves right. Test the vertex with $x = 2$ .

The short version

f (x) + k

moves up,

f (x - h)

moves right,

- f (x)

flips over the

x

-axis, and

a f (x)

stretches vertically.

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