To find the vertex of the quadratic function [tex]\( y = -(x-1)(x+3) \)[/tex], we follow these steps:
1. Expand the given function:
[tex]\[
y = -(x-1)(x+3)
\][/tex]
First, let's expand the expression inside the parentheses:
[tex]\[
(x-1)(x+3) = x^2 + 3x - x - 3 = x^2 + 2x - 3
\][/tex]
Now, multiplying by [tex]\(-1\)[/tex]:
[tex]\[
y = -(x^2 + 2x - 3) = -x^2 - 2x + 3
\][/tex]
So, we have the function in standard form:
[tex]\[
y = -x^2 - 2x + 3
\][/tex]
2. Identify the coefficients:
In the standard form [tex]\( y = ax^2 + bx + c \)[/tex], the coefficients are:
[tex]\[
a = -1, \quad b = -2, \quad c = 3
\][/tex]
3. Find the vertex:
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] is given by the coordinates [tex]\((h, k)\)[/tex], where:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
h = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1
\][/tex]
To find the y-coordinate [tex]\(k\)[/tex], substitute [tex]\(h = -1\)[/tex] back into the function:
[tex]\[
k = a(-1)^2 + b(-1) + c
\][/tex]
[tex]\[
k = -1(-1)^2 - 2(-1) + 3 = -1(1) + 2 + 3 = -1 + 2 + 3 = 4
\][/tex]
Thus, the vertex of the function is:
[tex]\[
(-1, 4)
\][/tex]
4. Conclusion:
The vertex of the graph of the function [tex]\( y = -(x-1)(x+3) \)[/tex] is [tex]\(\boxed{(-1, 4)}\)[/tex].
