Question

5
match the correct behavior with the function.
function
$f(x) = 2|x - 6| - 5$
$g(x) = -dfrac{1}{2}|x + 3| + 2$
$h(x) = -4|x + 6| + 1$

opens up
opens down
vertex moves to the left
vertex moves to the right
vertex moves up
vertex moves down
vsf is a stretch (more steep) than the parent function
vsf is a compression (less steep) than the parent function

Explanation:

Step1: Recall absolute value form

The parent function is $f(x)=|x|$. Transformed form: $a|x-h|+k$, where:

• $a>0$: opens up; $a<0$: opens down
• $h$: right if $h>0$, left if $h<0$
• $k$: up if $k>0$, down if $k<0$
• $|a|>1$: stretch; $0<|a|<1$: compression

Step2: Analyze $f(x)=2|x-6|-5$

• $a=2>0$: opens up
• $h=6>0$: vertex right
• $k=-5<0$: vertex down
• $|a|=2>1$: stretch

<Expression>
Opens Up: ✅, Vertex Moves to the RIGHT: ✅, Vertex Moves Down: ✅, VSF is a stretch (more steep) than the parent function: ✅
</Expression>

Step3: Analyze $g(x)=-\frac{1}{2}|x+3|+2$

• $a=-\frac{1}{2}<0$: opens down
• $h=-3<0$: vertex left
• $k=2>0$: vertex up
• $|a|=\frac{1}{2}<1$: compression

<Expression>
Opens Down: ✅, Vertex Moves to the LEFT: ✅, Vertex Moves Up: ✅, VSF is a compression (less steep) that the parent function: ✅
</Expression>

Step4: Analyze $h(x)=-4|x+6|+1$

• $a=-4<0$: opens down
• $h=-6<0$: vertex left
• $k=1>0$: vertex up
• $|a|=4>1$: stretch

<Expression>
Opens Down: ✅, Vertex Moves to the LEFT: ✅, Vertex Moves Up: ✅, VSF is a stretch (more steep) than the parent function: ✅
</Expression>

Answer:

Function	Opens Up	Opens Down	Vertex Moves to the LEFT	Vertex Moves to the RIGHT	Vertex Moves Up	Vertex Moves Down	VSF is a stretch (more steep) than the parent function	VSF is a compression (less steep) that the parent function
$g(x)=-\frac{1}{2}	x+3	+2$	❌	✅	✅	❌	✅	❌	❌	✅
$h(x)=-4	x+6	+1$	❌	✅	✅	❌	✅	❌	✅	❌