We can always locate the position of an absolute value function if we identify the transformations being applied to the parent function

f (x) = ∣ x ∣ .

To do so, let's first consider the general form of a transformed absolute value function.

y = a ∣ x - h ∣ + k

In this general form,

∣ a ∣

is the stretch or compression factor,

h

is a horizontal translation, and

k

is a vertical translation. Because translations shift an entire function, the vertex of a translated absolute value function is located at

(h, k)

and the axis of symmetry is the line

x = h .

Now, we will analyze an example.

g (x) = ∣ x + 2 ∣ - 3 ⇓ g (x) = 1 ∣ x - (- 2) ∣ + (- 3)

In this case, we have a vertex of

(- 2, - 3)

and an axis of symmetry of

x = - 2 .

Next, let's look at all of the possible transformations so that we can more clearly identify what is happening to our function.

Transformations of $y = ∣ x ∣$
Vertical Translations	$Translation up k units, k > 0 y = ∣ x ∣ + k$
Vertical Translations	$Translation down k units, k > 0 y = ∣ x ∣ - k$
Horizontal Translations	$Translation right h units, h > 0 y = ∣ x - h ∣$
Horizontal Translations	$Translation left h units, h > 0 y = ∣ x + h ∣$
Vertical Stretch or Compression	$Vertical stretch, a > 1 y = a ∣ x ∣$
Vertical Stretch or Compression	$Vertical compression, 0 < a < 1 y = a ∣ x ∣$
Horizontal Stretch or Compression	$Horizontal stretch, 0 < b < 1 y = ∣ b x ∣$
Horizontal Stretch or Compression	$Horizontal compression, b > 1 y = ∣ b x ∣$
Reflections	$In the x - axis y = - ∣ x ∣$
Reflections	$In the y - axis y = ∣ - x ∣$

Using the table, we can now describe the transformations.

A horizontal translation to the left by $2$ units.
A vertical translation down by $3$ units.

To graph the function, recall that the the parent function $f (x) = ∣ x ∣$ has a vertex in the origin $(0, 0),$ it is V-shaped and symmetric across the $y -$ axis.

Now, we will apply the described transformations.

Notice that this is only an example. There are infinitely many different ways to transform the parent function, but as long as we can identify each of the transformations applied we will be able to find the position of the new function.

Exercise

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