Question

complete the statements for the function ( y = \frac{1}{2}|x + 2| - 1 ).

the function is positive when (\boxed{quad}).
the function is negative when (\boxed{quad}).
the function is increasing when (\boxed{quad}).
the function is decreasing when (\boxed{quad}).

options: ( x < -2 ), ( x > -2 ), ( x < -4 ) and ( x > 0 ), ( -4 < x < 0 ), ( x < -1 ), ( x > -1 )

Explanation:

Step1: Find when function is positive

Set $\frac{1}{2}|x+2| - 1 > 0$
$\frac{1}{2}|x+2| > 1$
$|x+2| > 2$
This splits into two inequalities:
$x+2 > 2 \implies x > 0$
$x+2 < -2 \implies x < -4$
So $x < -4$ and $x > 0$.

Step2: Find when function is negative

Set $\frac{1}{2}|x+2| - 1 < 0$
$\frac{1}{2}|x+2| < 1$
$|x+2| < 2$
This gives:
$-2 < x+2 < 2$
$-4 < x < 0$

Step3: Find increasing interval

The vertex of $y=\frac{1}{2}|x+2|-1$ is at $x=-2$. For absolute value functions with positive coefficient on the absolute value, the function increases to the right of the vertex. So $x > -2$.

Step4: Find decreasing interval

The function decreases to the left of the vertex $x=-2$. So $x < -2$.

Answer:

The function is positive when $\boldsymbol{x < -4}$ and $\boldsymbol{x > 0}$.
The function is negative when $\boldsymbol{-4 < x < 0}$.
The function is increasing when $\boldsymbol{x > -2}$.
The function is decreasing when $\boldsymbol{x < -2}$.