Question

complete the statements for the function $y = \frac{1}{2}|x + 2| - 1$. the function is positive when \boxed{}. the function is negative when \boxed{}. the function is increasing when \boxed{}. the function is decreasing when \boxed{}. options: $x > -2$, $-4 < x < 0$, $x > -1$, $x < -4$ and $x > 0$, $x < -1$, $x < -2$

Explanation:

Step1: Find when the function is positive

To find when \( y=\frac{1}{2}|x + 2|-1>0\), we solve the inequality:
\[
\frac{1}{2}|x + 2|-1>0
\]
Add 1 to both sides:
\[
\frac{1}{2}|x + 2|>1
\]
Multiply both sides by 2:
\[
|x + 2|>2
\]
This gives two cases:
Case 1: \(x + 2>2\) which simplifies to \(x>0\)
Case 2: \(x + 2<-2\) which simplifies to \(x<-4\)
So the function is positive when \(x < - 4\) and \(x>0\)

Step2: Find when the function is negative

To find when \( y=\frac{1}{2}|x + 2|-1<0\), we solve the inequality:
\[
\frac{1}{2}|x + 2|-1<0
\]
Add 1 to both sides:
\[
\frac{1}{2}|x + 2|<1
\]
Multiply both sides by 2:
\[
|x + 2|<2
\]
This gives \(-2Subtract 2 from all parts: \(-4So the function is negative when \(-4 < x<0\)

Step3: Find when the function is increasing

The vertex of the absolute - value function \(y = a|x - h|+k\) is at \((h,k)\). For \(y=\frac{1}{2}|x + 2|-1\), the vertex is at \((-2,-1)\). The slope of the right - hand side (when \(x\geq - 2\)) of the absolute - value function \(y=\frac{1}{2}|x + 2|-1\) is positive (\(a=\frac{1}{2}>0\)). So the function is increasing when \(x>-2\)

Step4: Find when the function is decreasing

The slope of the left - hand side (when \(x < - 2\)) of the absolute - value function \(y=\frac{1}{2}|x + 2|-1\) is negative. So the function is decreasing when \(x<-2\)

Answer:

The function is positive when \(x < - 4\) and \(x>0\). The function is negative when \(-4 < x<0\). The function is increasing when \(x>-2\). The function is decreasing when \(x<-2\)

So the answers are:

• The function is positive when \(x < - 4\) and \(x>0\)
• The function is negative when \(-4 < x<0\)
• The function is increasing when \(x>-2\)
• The function is decreasing when \(x<-2\)