We are asked to write the absolute value function

y = - ∣ x - 3 ∣ + 2

as a piecewise function. Let's start by recalling that how we write the vertex form of an absolute value function,

g (x) = a ∣ x - h ∣ + k,

as a piecewise function.

g (x) = {a (- (x - h)) + k, if x - h < 0 a (x - h) + k, if x - h \geq 0

Using this, let's identify the values for our function.

g (x) = y = a ∣ x - h ∣ + k ⇓ - 1 ∣ x - 3 ∣ + 2

Now, we will substitute

a = - 1,

h = 3,

and

k = 2

into the above piecewise function.

{a (- (x - h)) + k, if x - h < 0 a (x - h) + k, if x - h \geq 0

$(I), (II):$ Substitute values

{- 1 (- (x - 3)) + 2, if x - 3 < 0 - 1 (x - 3) + 2, if x - 3 \geq 0

▼

(I), (II):

Simplify

Distr

$(I):$ Distribute $- 1$

{- 1 (- x + 3) + 2, if x - 3 < 0 - 1 (x - 3) + 2, if x - 3 \geq 0

$(I), (II):$ Distribute $- 1$

{x - 3 + 2, if x - 3 < 0 - x + 3 + 2, if x - 3 \geq 0

$(I), (II):$ Add and subtract terms

{x - 1, if x - 3 < 0 - x + 5, if x - 3 \geq 0

Great! Last, let's rearrange the domain of this piecewise function by adding

3

to both sides of the inequalities.

y = {- x - 1, - x + 5, if x < 3 if x \geq 3

Alternative Solution

Using Graphs

This time we will begin with drawing the graph of our given function.

As we can see, the behavior of the absolute value function changes at the vertex. Therefore, this is a good point to separate the graph into two pieces, both being straight lines. Our next step will then be to find the equations for both lines. For this, let's recall the slope-intercept form of an equation.

y = m x + b

In this form,

m

is the slope of the line and

b

is the

y -

intercept. Now, we can start with the line to the left of the vertex first. We can identify the

y -

intercept and slope from the graph shown below.

For this line, the $y -$ intercept is $b = - 1$ and the slope is $m = 1 .$ Therefore, the equation for it is $y = x - 1 .$ We will now do the same with the line to the right of the vertex.

Then, for this second line, the

y -

intercept is

b = 5

and the slope is

m = - 1 .

Therefore, the equation for it is

y = - x + 5 .

Knowing the equations for both lines, we can now write the absolute value function

- ∣ x - 3 ∣ + 2

as the combination of two functions.

y = {- x - 1 - x + 5

Last, let's decide the domain of each piece. The pieces meet and change directions at

x = 3,

so this is where our domain needs to change. The point at

x = 3

can belong to either piece, as long as it belongs to only one of them. This means that we can correctly write the piecewise function in two different ways.

y = {- x - 1, - x + 5, if x \leq 3 if x > 3 or y = {- x - 1, - x + 5, if x < 3 if x \geq 3

Exercise

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