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Answer :
To determine which equation represents the absolute value function with a vertex at [tex]\((1, 3)\)[/tex], we need to use the standard form of an absolute value function equation:
[tex]\[ g(x) = a|x - h| + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the graph.
Given that the vertex is [tex]\((1, 3)\)[/tex]:
1. Identify the vertex:
[tex]\((h, k) = (1, 3)\)[/tex].
2. Substitute the vertex into the equation:
- Replace [tex]\(h\)[/tex] with [tex]\(1\)[/tex].
- Replace [tex]\(k\)[/tex] with [tex]\(3\)[/tex].
This gives us the equation:
[tex]\[ g(x) = |x - 1| + 3 \][/tex]
3. Compare with the given options:
- [tex]\(g(x) = |x+1|+3\)[/tex]
- [tex]\(g(x) = |x+3|-1\)[/tex]
- [tex]\(g(x) = |x-1|+3\)[/tex]
- [tex]\(g(x) = |x+3|+1\)[/tex]
4. Choose the correct equation:
The correct equation is [tex]\(g(x) = |x-1|+3\)[/tex], which matches the vertex [tex]\((1, 3)\)[/tex].
So, the equation that represents the function graphed on the coordinate plane with a vertex at [tex]\((1, 3)\)[/tex] is:
[tex]\[ g(x) = |x-1|+3 \][/tex]
[tex]\[ g(x) = a|x - h| + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the graph.
Given that the vertex is [tex]\((1, 3)\)[/tex]:
1. Identify the vertex:
[tex]\((h, k) = (1, 3)\)[/tex].
2. Substitute the vertex into the equation:
- Replace [tex]\(h\)[/tex] with [tex]\(1\)[/tex].
- Replace [tex]\(k\)[/tex] with [tex]\(3\)[/tex].
This gives us the equation:
[tex]\[ g(x) = |x - 1| + 3 \][/tex]
3. Compare with the given options:
- [tex]\(g(x) = |x+1|+3\)[/tex]
- [tex]\(g(x) = |x+3|-1\)[/tex]
- [tex]\(g(x) = |x-1|+3\)[/tex]
- [tex]\(g(x) = |x+3|+1\)[/tex]
4. Choose the correct equation:
The correct equation is [tex]\(g(x) = |x-1|+3\)[/tex], which matches the vertex [tex]\((1, 3)\)[/tex].
So, the equation that represents the function graphed on the coordinate plane with a vertex at [tex]\((1, 3)\)[/tex] is:
[tex]\[ g(x) = |x-1|+3 \][/tex]
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