To determine the equation of the absolute value function that has a vertex at [tex]\((2, 3)\)[/tex] and crosses the [tex]\(x\)[/tex]-axis at the points [tex]\((-1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex], follow these steps:
1. Form of the Absolute Value Function:
The general form for an absolute value function with a vertex [tex]\((h, k)\)[/tex] is:
[tex]\[
y = a|x - h| + k
\][/tex]
Here, the vertex is [tex]\((2, 3)\)[/tex], so the form of the function is:
[tex]\[
y = a|x - 2| + 3
\][/tex]
2. Determining the Value of [tex]\(a\)[/tex]:
Given that the function crosses the [tex]\(x\)[/tex]-axis at points where [tex]\(y = 0\)[/tex], we can use these points to find the value of [tex]\(a\)[/tex].
1. Substitute the crossing point [tex]\((-1, 0)\)[/tex] into the equation:
[tex]\[
0 = a|(-1) - 2| + 3
\][/tex]
Simplify the absolute value:
[tex]\[
0 = a| -3 | + 3
\][/tex]
[tex]\[
0 = 3a + 3
\][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[
0 - 3 = 3a
\][/tex]
[tex]\[
-3 = 3a
\][/tex]
[tex]\[
a = -1
\][/tex]
3. Substitute the Value of [tex]\(a\)[/tex] into the Equation:
Now that we have [tex]\(a = -1\)[/tex], substitute this back into the general form of the function:
[tex]\[
y = -|x - 2| + 3
\][/tex]
4. Equation When [tex]\(y = 0\)[/tex]:
Given [tex]\(y = 0\)[/tex], the equation becomes:
[tex]\[
0 = -|x - 2| + 3
\][/tex]
Thus, the equation for the given absolute value function when [tex]\(y = 0\)[/tex] is:
[tex]\[
D. \, 0 = -|x - 2| + 3
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
