To find the coordinates of the vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex], we can use the vertex formula. The general form of a parabolic equation is [tex]\( y = ax^2 + bx + c \)[/tex].
The vertex [tex]\((h, k)\)[/tex] of a parabola can be found using the following steps:
1. Find the x-coordinate of the vertex, [tex]\(x = h\)[/tex]:
The formula to find the x-coordinate of the vertex is:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
In the given equation [tex]\( y = 5x^2 - 70x + 230 \)[/tex]:
[tex]\[
a = 5, \quad b = -70, \quad c = 230
\][/tex]
Plugging in the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x = -\frac{-70}{2 \times 5} = \frac{70}{10} = 7
\][/tex]
2. Find the y-coordinate of the vertex, [tex]\(y = k\)[/tex]:
Substitute [tex]\(x = 7\)[/tex] back into the original equation to find [tex]\(y\)[/tex]:
[tex]\[
y = 5(7)^2 - 70(7) + 230
\][/tex]
Calculate each term:
[tex]\[
5(7)^2 = 5 \times 49 = 245
\][/tex]
[tex]\[
-70(7) = -490
\][/tex]
Now sum these results with the constant term:
[tex]\[
y = 245 - 490 + 230
\][/tex]
Simplify the expression:
[tex]\[
y = -245 + 230 = -15
\][/tex]
So the coordinates of the vertex of the parabola [tex]\( y = 5x^2 - 70x + 230 \)[/tex] are [tex]\( (7, -15) \)[/tex].
Therefore, the vertex is:
[tex]\[
(7, -15)
\][/tex]
