Given the problem: The vertex of the parabola is at [tex]\((-3, -2)\)[/tex]. Which of the following could be its equation?
To solve this problem, we need to recall the vertex form of a parabola's equation, which is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
In this problem, the vertex is given as [tex]\((-3, -2)\)[/tex]. Therefore, our vertex form equation needs to fit:
[tex]\[ y = a(x - (-3))^2 - 2 \][/tex]
or more simply,
[tex]\[ y = a(x + 3)^2 - 2 \][/tex]
Let's analyze each given option:
Option A: [tex]\( y = -2(x - 3)^2 + 2 \)[/tex]
- Here, [tex]\(h = 3\)[/tex] and [tex]\(k = 2\)[/tex]
- The vertex would be at [tex]\((3, 2)\)[/tex], which does not match the given vertex [tex]\((-3, -2)\)[/tex].
Option B: [tex]\( y = -2(x + 3)^2 + 2 \)[/tex]
- Here, [tex]\(h = -3\)[/tex] and [tex]\(k = 2\)[/tex]
- The vertex would be at [tex]\((-3, 2)\)[/tex], which does not match the given vertex [tex]\((-3, -2)\)[/tex].
Option C: [tex]\( y = -2(x + 3)^2 - 2 \)[/tex]
- Here, [tex]\(h = -3\)[/tex] and [tex]\(k = -2\)[/tex]
- The vertex would be at [tex]\((-3, -2)\)[/tex], which matches the given vertex [tex]\((-3, -2)\)[/tex].
Option D: [tex]\( y = -2(x - 3)^2 - 2 \)[/tex]
- Here, [tex]\(h = 3\)[/tex] and [tex]\(k = -2\)[/tex]
- The vertex would be at [tex]\((3, -2)\)[/tex], which does not match the given vertex [tex]\((-3, -2)\)[/tex].
Based on this analysis, the equation that fits the given vertex [tex]\((-3, -2)\)[/tex] is:
[tex]\[ \boxed{C. \, y = -2(x + 3)^2 - 2} \][/tex]
