To determine which of the given equations represents a parabola with a vertex at [tex]\((3,0)\)[/tex], let's analyze the vertex form of a parabola's equation.
The vertex form of a parabola's equation is:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h,k)\)[/tex] represents the vertex of the parabola.
Given that the vertex is [tex]\((3,0)\)[/tex], we substitute [tex]\(h = 3\)[/tex] and [tex]\(k = 0\)[/tex] into the vertex form:
[tex]\[ y = a(x - 3)^2 + 0 \][/tex]
[tex]\[ y = a(x - 3)^2 \][/tex]
Among the given options, we need to find the equation that matches this form.
Let's examine each option:
1. [tex]\( y = x^2 + 3 \)[/tex]
- This is in the standard form [tex]\( y = ax^2 + bx + c \)[/tex] and has a vertex at [tex]\((0,3)\)[/tex], not [tex]\((3,0)\)[/tex].
2. [tex]\( y = x^2 - 3 \)[/tex]
- This is in the standard form [tex]\( y = ax^2 + bx + c \)[/tex] and has a vertex at [tex]\((0,-3)\)[/tex], not [tex]\((3,0)\)[/tex].
3. [tex]\( y = (x + 3)^2 \)[/tex]
- This is in the vertex form [tex]\( y = a(x - (-3))^2 + 0 \)[/tex] and has a vertex at [tex]\((-3,0)\)[/tex], not [tex]\((3,0)\)[/tex].
4. [tex]\( y = (x - 3)^2 \)[/tex]
- This is in the vertex form [tex]\( y = a(x - 3)^2 + 0 \)[/tex] and has a vertex at [tex]\((3,0)\)[/tex], which matches the given vertex.
Based on our analysis, the correct equation for a parabola with a vertex at [tex]\((3,0)\)[/tex] is:
[tex]\[ y = (x - 3)^2 \][/tex]
Therefore, the correct answer is:
[tex]\[ 4 \][/tex]
