To analyze the quadratic function [tex]\( y = -x^2 + 8x + 2 \)[/tex], we'll find both its vertex and its zeros (roots).
### Finding the Vertex
The vertex form of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] has its vertex at [tex]\( x = -\frac{b}{2a} \)[/tex].
For the given function:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = 2 \)[/tex]
We calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} = -\frac{8}{2 \cdot (-1)} = \frac{8}{2} = 4 \][/tex]
To find the y-coordinate of the vertex, we substitute [tex]\( x = 4 \)[/tex] back into the original equation:
[tex]\[ y = -4^2 + 8 \cdot 4 + 2 \][/tex]
[tex]\[ y = -16 + 32 + 2 \][/tex]
[tex]\[ y = 18 \][/tex]
Thus, the vertex of the quadratic function is [tex]\((4, 18)\)[/tex].
### Finding the Zeros
To find the zeros of the function, we solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = -x^2 + 8x + 2 \][/tex]
We use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] to find the roots.
Plugging in the coefficients [tex]\( a = -1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 2 \)[/tex]:
1. Compute the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = 8^2 - 4 \cdot (-1) \cdot 2 = 64 + 8 = 72 \][/tex]
2. Compute the roots:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a} = \frac{-8 \pm \sqrt{72}}{2 \cdot (-1)} = \frac{-8 \pm 6\sqrt{2}}{-2}
\][/tex]
Separating the plus and minus cases:
[tex]\[
x_1 = \frac{-8 + 6\sqrt{2}}{-2} = \frac{8 - 6 \sqrt{2}}{2} = -0.2
\][/tex]
[tex]\[
x_2 = \frac{-8 - 6\sqrt{2}}{-2} = \frac{8 + 6\sqrt{2}}{2} = 8.2
\][/tex]
Therefore, the zeros (roots) of the quadratic function, sorted from least to greatest, are approximately [tex]\( -0.2 \)[/tex] and [tex]\( 8.2 \)[/tex].
### Summarized Results
- Vertex: [tex]\( (4, 18) \)[/tex]
- Zeros: [tex]\( -0.2 \)[/tex] and [tex]\( 8.2 \)[/tex]
Completing the given blanks, the final solution is:
Vertex: [tex]\((4, 18)\)[/tex]
Zeros: [tex]\((-0.2, 8.2)\)[/tex]
In symbols, it looks like this:
```
Vertex: ( 4 , 18 )
Solutions: ( -0.2 , 8.2 )
```
