Question

function
transform
$g(x) = -|x+2| - 4$
$g(x) = -|x| - 4$
$g(x) = |x+4| + 2$
$g(x) = -3|x - 4| + 2$

Explanation:

Step1: Analyze the vertex of the graph

The graph is a V - shaped graph (absolute value function graph) with a vertex. From the graph, we can see that the vertex of the absolute - value function \(y = a|x - h|+k\) has coordinates \((h,k)\). Looking at the graph, the vertex seems to be at \((4,2)\)? Wait, no, let's re - examine. Wait, the graph has a peak (vertex) and we can also check the slope and the vertex form. Let's recall the vertex form of an absolute value function \(g(x)=a|x - h|+k\), where \((h,k)\) is the vertex.

Looking at the graph, when \(x = 4\), let's check the value. Also, the graph is opening downwards, so \(a<0\). Let's check each function:

1. For \(g(x)=-|x + 2|-4\): The vertex is at \((-2,-4)\), and the graph opens down. But our graph's vertex is not at \((-2,-4)\).
2. For \(g(x)=-|x|-4\): The vertex is at \((0,-4)\), graph opens down. Not matching.
3. For \(g(x)=|x + 4|+2\): The vertex is at \((-4,2)\), graph opens up (\(a = 1>0\)), but our graph opens down.
4. For \(g(x)=-3|x - 4|+2\): The vertex is at \((4,2)\), and \(a=-3<0\) (opens down). Let's check the slope. The general form of \(y=a|x - h|+k\), for \(x>h\), \(y=a(x - h)+k\), slope is \(a\); for \(x

Let's check the y - intercept. When \(x = 0\), \(g(0)=-3|0 - 4|+2=-3\times4 + 2=-12 + 2=-10\), which matches the y - intercept of the graph (when \(x = 0\), \(y=-10\)). Also, the vertex at \((4,2)\) seems to match the peak of the graph. And since \(a=-3<0\), the graph opens down, which matches the graph's shape (opening downwards).

Step2: Verify the function with the graph's key points

We can also check another point. Let's take \(x = 6\). For \(g(x)=-3|x - 4|+2\), when \(x = 6\), \(g(6)=-3|6 - 4|+2=-3\times2+2=-6 + 2=-4\). Looking at the graph, when \(x = 6\), the y - value is around - 4, which matches.

For the other functions:

• For \(g(x)=-|x + 2|-4\), when \(x = 0\), \(g(0)=-|0 + 2|-4=-2-4=-6

eq - 10\)

• For \(g(x)=-|x|-4\), when \(x = 0\), \(g(0)=-0 - 4=-4

eq - 10\)

• For \(g(x)=|x + 4|+2\), when \(x = 0\), \(g(0)=|0 + 4|+2=4 + 2=6

eq - 10\)

Answer:

\(g(x)=-3|x - 4|+2\)