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In this section, all the transformations will be presented together.

It is important to be able to identify how each parameter in the vertex form equation affects the graph.

y = a|x-h| + k, where a represents a reflection across the x-axis and/or a vertical stretch or compression, k represents a vertical translation, and h represents a horizontal translation

Example

How is the graph f(x) = 2 | x + 3 | - 1 translated from the parent function y = | x |?

a = 2 is a reflection across the x-axis, a stretch creating a narrow graph
h = -3 is a horizontal translation, moving the graph to the left three units
k = -1 is a vertical translation, moving the graph down one unit

If the graph of f(x) = 2 | x + 3 | - 1 is translated 5 units up, what is the new equation?

A translation of 5 units up is vertical and affects ONLY the “k” in the equation.

If f(x) = 2 | x + 3 | - 1 ⇒ k = -1.

To translate this graph up, you should add to “k”.

Starting at -1 and moving up 5 units, gives you -1 + 5 = 4, so the new h = 4.

The new equation is: f(x) = 2 | x + 3 | + 4

The new graph is:

graphing calculator screen showing f(x) = 2 | x+3 |+4 and f(x) = 2 | x+3 |-1

The tables of values for the two equations are:

f(x) = 2 | x + 3 | − 1

x	y
0	5
-1	3
-3	-1
-5	3
-6	5

and

**Vertex

f(x) = 2 | x + 3 | + 4

x	y
0	10
-1	8
-3	4
-5	8
-6	10

**Vertex