Certainly! Let's analyze the problem step-by-step.
We are given that the vertex of a parabola is at [tex]\((-3, 2)\)[/tex]. The general form for a parabola's vertex form is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Here, [tex]\( h = -3 \)[/tex] and [tex]\( k = 2 \)[/tex].
Substituting the vertex into the formula, we get:
[tex]\[ y = a(x + 3)^2 + 2 \][/tex]
Now, we need to identify which of the given options match this form. Let’s examine each option:
Option A: [tex]\( y = 4(x - 3)^2 + 2 \)[/tex]
- Here, the equation is [tex]\( y = 4(x - 3)^2 + 2 \)[/tex]. The vertex for this form is [tex]\( (3, 2) \)[/tex], which does not match our given vertex [tex]\((-3, 2)\)[/tex]. Therefore, this option is incorrect.
Option B: [tex]\( y = 4(x - 3)^2 - 2 \)[/tex]
- Here, the equation is [tex]\( y = 4(x - 3)^2 - 2 \)[/tex]. The vertex for this form is [tex]\( (3, -2) \)[/tex], which also does not match our given vertex [tex]\((-3, 2)\)[/tex]. Therefore, this option is incorrect.
Option C: [tex]\( y = 4(x + 3)^2 + 2 \)[/tex]
- Here, the equation is [tex]\( y = 4(x + 3)^2 + 2 \)[/tex]. The vertex for this form is [tex]\((-3, 2) \)[/tex], which matches our given vertex. Therefore, this option is correct.
Option D: [tex]\( y = 4(x + 3)^2 + 2 \)[/tex]
- Here, the equation is [tex]\( y = 4(x + 3)^2 + 2 \)[/tex]. The vertex for this form is also [tex]\((-3, 2) \)[/tex], which matches our given vertex. Therefore, this option is correct as well.
In conclusion, the equations that could correspond to a parabola with the vertex at [tex]\((-3, 2)\)[/tex] are given in options [tex]\(C\)[/tex] and [tex]\(D\)[/tex].
