To find the coordinates of the vertex for the quadratic function [tex]\( f(x) = -x^2 - 8x + 4 \)[/tex], we need to follow these steps:
1. Identify the coefficients: The quadratic function is generally in the form [tex]\( ax^2 + bx + c \)[/tex]. Here, we have:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 4 \)[/tex]
2. Calculate the x-coordinate of the vertex: The formula to find the x-coordinate of the vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
x = -\frac{-8}{2 \cdot (-1)} = \frac{8}{-2} = -4
\][/tex]
4. Calculate the y-coordinate of the vertex: To find the y-coordinate, substitute [tex]\( x = -4 \)[/tex] back into the original quadratic function [tex]\( f(x) \)[/tex]:
[tex]\[
f(-4) = -(-4)^2 - 8(-4) + 4
\][/tex]
[tex]\[
f(-4) = -(16) + 32 + 4
\][/tex]
[tex]\[
f(-4) = -16 + 32 + 4 = 20
\][/tex]
5. Write the coordinates of the vertex: Combining the x-coordinate and y-coordinate, we get the vertex of the parabola.
The vertex is [tex]\((-4, 20)\)[/tex].
