To determine the correct equation of a parabola with a vertex at [tex]\((-2, 5)\)[/tex], we can use the vertex form of a parabola's equation:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Given that the vertex is [tex]\((-2, 5)\)[/tex]:
- [tex]\(h = -2\)[/tex]
- [tex]\(k = 5\)[/tex]
Substituting these values into the vertex form, we get:
[tex]\[ y = a(x - (-2))^2 + 5 \][/tex]
[tex]\[ y = a(x + 2)^2 + 5 \][/tex]
Now, let's analyze each given option to see which one matches this form:
- Option A: [tex]\(y = 3(x + 2)^2 - 5\)[/tex]
In this case, [tex]\(k = -5\)[/tex], which does not match our vertex form where [tex]\(k = 5\)[/tex]. So, this is incorrect.
- Option B: [tex]\(y = 3(x - 2)^2 + 5\)[/tex]
In this case, [tex]\(h = 2\)[/tex], which does not match our vertex form where [tex]\(h = -2\)[/tex]. So, this is incorrect.
- Option C: [tex]\(y = 3(x - 2)^2 - 5\)[/tex]
In this case, both [tex]\(h = 2\)[/tex] and [tex]\(k = -5\)[/tex] are incorrect. Thus, this option does not match the given vertex. So, this is incorrect.
- Option D: [tex]\(y = 3(x + 2)^2 + 5\)[/tex]
Here, [tex]\(h = -2\)[/tex] and [tex]\(k = 5\)[/tex], which exactly match the values in our vertex form. Therefore, this is the correct option.
Based on the analysis, the correct equation of the parabola with vertex [tex]\((-2, 5)\)[/tex] is:
[tex]\[ \boxed{y = 3(x + 2)^2 + 5} \][/tex]
