To determine the correct equation for the absolute value function with the given vertex and x-intercepts, let's break down the problem step by step.
### Step 1: Identify the Vertex of the Absolute Value Function
The vertex of the absolute value function is given as [tex]\((2, 3)\)[/tex]. The general form of an absolute value function is:
[tex]\[ y = a |x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex. For our problem, [tex]\(h = 2\)[/tex] and [tex]\(k = 3\)[/tex]. Therefore, the equation can start as:
[tex]\[ y = a |x - 2| + 3 \][/tex]
### Step 2: Determine the x-Intercepts and Verify the Function
The x-intercepts of the function are [tex]\((-1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex]. This means that when [tex]\(y = 0\)[/tex], the equation should satisfy:
[tex]\[ 0 = a |x - 2| + 3 \][/tex]
### Step 3: Solve for the Absolute Value at the Intercepts
Now, let's substitute the x-intercepts into the equation and ensure it holds true:
1. Substituting [tex]\( x = -1 \)[/tex]:
[tex]\[ 0 = a |-1 - 2| + 3 \][/tex]
[tex]\[ 0 = a | -3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
To satisfy this equation, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
2. Substituting [tex]\( x = 5 \)[/tex]:
[tex]\[ 0 = a |5 - 2| + 3 \][/tex]
[tex]\[ 0 = a | 3 | + 3 \][/tex]
[tex]\[ 0 = 3a + 3 \][/tex]
Again, [tex]\(a\)[/tex] must be [tex]\(-1\)[/tex].
### Step 4: Verify Both Conditions
Given that both conditions for the x-intercepts hold true with [tex]\(a = 1\)[/tex], we can conclude that the correct equation fits the form:
[tex]\[ y = |x - 2| + 3 \][/tex]
### Step 5: Check the Given Options
Now let's verify which option matches our derived equation when [tex]\(y = 0\)[/tex]:
A. [tex]\(0 = |x + 2| + 3\)[/tex]
B. [tex]\(0 = |x - 2| + 3\)[/tex]
C. [tex]\(0 = -|x + 2| + 3\)[/tex]
Option B matches the form [tex]\(0 = |x - 2| + 3\)[/tex] and fits within our calculations.
### Conclusion
Therefore, the correct equation for the absolute value function, when [tex]\(y = 0\)[/tex], is:
[tex]\[ \boxed{0 = |x - 2| + 3} \][/tex]
