Question

1. $g(x) = |x - 4| + 2$
2. $g(x) = |x + 7| - 1$
3. $g(x) = |x + 5| - 3$

Explanation:

For $g(x) = |x-4| + 2$

Step1: Find vertex of the function

The vertex form of an absolute value function is $g(x)=|x-h|+k$, where $(h,k)$ is the vertex. Here $h=4$, $k=2$, so vertex is $(4, 2)$.

Step2: Find left side points

For $x<4$, use $g(x)=-(x-4)+2=-x+6$.

• $x=0$: $g(0)=-0+6=6$, point $(0,6)$
• $x=-4$: $g(-4)=-(-4)+6=10$, point $(-4,10)$

Step3: Find right side points

For $x>4$, use $g(x)=(x-4)+2=x-2$.

• $x=8$: $g(8)=8-2=6$, point $(8,6)$
• $x=6$: $g(6)=6-2=4$, point $(6,4)$

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For $g(x) = |x+7| - 1$

Step1: Find vertex of the function

Rewrite as $g(x)=|x-(-7)|+(-1)$, so vertex is $(-7, -1)$.

Step2: Find left side points

For $x<-7$, use $g(x)=-(x+7)-1=-x-8$.

• $x=-8$: $g(-8)=-(-8)-8=0$, point $(-8,0)$
• $x=-10$ (off grid, use $x=-6$ for right):

Step3: Find right side points

For $x>-7$, use $g(x)=(x+7)-1=x+6$.

• $x=0$: $g(0)=0+6=6$, point $(0,6)$
• $x=-4$: $g(-4)=-4+6=2$, point $(-4,2)$

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For $g(x) = |x+5| - 3$

Step1: Find vertex of the function

Rewrite as $g(x)=|x-(-5)|+(-3)$, so vertex is $(-5, -3)$.

Step2: Find left side points

For $x<-5$, use $g(x)=-(x+5)-3=-x-8$.

• $x=-8$: $g(-8)=-(-8)-8=0$, point $(-8,0)$
• $x=-6$: $g(-6)=-(-6)-8=-2$, point $(-6,-2)$

Step3: Find right side points

For $x>-5$, use $g(x)=(x+5)-3=x+2$.

• $x=0$: $g(0)=0+2=2$, point $(0,2)$
• $x=-4$: $g(-4)=-4+2=-2$, point $(-4,-2)$

Answer:

1. For $g(x)=|x-4|+2$:

• Plot vertex $(4,2)$, then points $(0,6)$, $(-4,10)$, $(8,6)$, $(6,4)$; connect to form a V-shape opening upward.

1. For $g(x)=|x+7|-1$:

• Plot vertex $(-7,-1)$, then points $(-8,0)$, $(0,6)$, $(-4,2)$; connect to form a V-shape opening upward.

1. For $g(x)=|x+5|-3$:

• Plot vertex $(-5,-3)$, then points $(-8,0)$, $(-6,-2)$, $(0,2)$, $(-4,-2)$; connect to form a V-shape opening upward.