Question

graph each absolute value function.

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

Explanation:

Step1: Find vertex of $g(x)=|x+1|+1$

The vertex form of an absolute value function is $g(x)=|x-h|+k$, where $(h,k)$ is the vertex. For $g(x)=|x+1|+1$, rewrite as $g(x)=|x-(-1)|+1$, so vertex is $(-1, 1)$.

Step2: Find 2 points for $g(x)=|x+1|+1$

When $x=0$: $g(0)=|0+1|+1=1+1=2$, so point $(0,2)$.
When $x=-2$: $g(-2)=|-2+1|+1=1+1=2$, so point $(-2,2)$.

Step3: Find vertex of $g(x)=|x-3|-5$

Using vertex form $g(x)=|x-h|+k$, here $h=3, k=-5$, so vertex is $(3, -5)$.

Step4: Find 2 points for $g(x)=|x-3|-5$

When $x=0$: $g(0)=|0-3|-5=3-5=-2$, so point $(0,-2)$.
When $x=6$: $g(6)=|6-3|-5=3-5=-2$, so point $(6,-2)$.

Step5: Plot and connect points

For $g(x)=|x+1|+1$: Plot vertex $(-1,1)$ and points $(0,2), (-2,2)$, then draw a V-shape through them.
For $g(x)=|x-3|-5$: Plot vertex $(3,-5)$ and points $(0,-2), (6,-2)$, then draw a V-shape through them.

Answer:

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

• Vertex at $(-1, 1)$, additional points $(0, 2)$ and $(-2, 2)$; graph is a V-shape opening upward with these points.

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

• Vertex at $(3, -5)$, additional points $(0, -2)$ and $(6, -2)$; graph is a V-shape opening upward with these points.