To find the vertex of the given parabola, you need to express the equation in the standard vertex form of a parabola, which is:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
The given equation is:
[tex]\[ y - 3 = \frac{1}{2}(x + 5)^2 \][/tex]
We can compare this with the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex] to identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
1. The equation [tex]\( y - 3 = \frac{1}{2}(x + 5)^2 \)[/tex] can be written as [tex]\( y = \frac{1}{2}(x + 5)^2 + 3 \)[/tex].
2. By comparing [tex]\( y = \frac{1}{2}(x + 5)^2 + 3 \)[/tex] with [tex]\( y = a(x - h)^2 + k \)[/tex]:
- The term inside the squared parenthesis [tex]\((x + 5)\)[/tex] indicates [tex]\((x - (-5))\)[/tex]. Therefore, [tex]\( h = -5 \)[/tex].
- The constant term outside the parenthesis is [tex]\( 3 \)[/tex], so [tex]\( k = 3 \)[/tex].
Thus, the vertex coordinates are:
[tex]\[ (h, k) = (-5, 3) \][/tex]
So the vertex of the parabola is at [tex]\((-5, 3)\)[/tex].
Enter your answer in the boxes:
[tex]\[
(-5, \boxed{3})
\][/tex]
