Let's solve the problem step-by-step using the details.
Given the quadratic function:
[tex]\[ y = -x^2 - 8x + 8 \][/tex]
### Step 1: Find the Vertex
A quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] has its vertex at the point [tex]\((h, k)\)[/tex], where:
[tex]\[ h = -\frac{b}{2a} \][/tex]
For our function:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
So, the x-coordinate of the vertex, [tex]\( h \)[/tex], is:
[tex]\[ h = -\frac{-8}{2(-1)} = \frac{8}{-2} = -4 \][/tex]
Next, we substitute [tex]\( x = -4 \)[/tex] back into the original equation to find the y-coordinate, [tex]\( k \)[/tex]:
[tex]\[ k = -(-4)^2 - 8(-4) + 8 \][/tex]
[tex]\[ k = -16 + 32 + 8 \][/tex]
[tex]\[ k = 24 \][/tex]
Therefore, the vertex of the function is:
[tex]\[ \text{Vertex} = (-4, 24) \][/tex]
### Step 2: Find the Roots (Zeros)
The roots or zeros of the quadratic function are the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex]. Setting the equation to zero:
[tex]\[ -x^2 - 8x + 8 = 0 \][/tex]
This can be solved using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our function, [tex]\( a = -1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 8 \)[/tex]:
[tex]\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(-1)(8)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 + 32}}{-2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{96}}{-2} \][/tex]
[tex]\[ x = \frac{8 \pm 4\sqrt{6}}{-2} \][/tex]
Thus, the two roots are:
[tex]\[ x_1 = \frac{8 + 4\sqrt{6}}{-2} \][/tex]
[tex]\[ x_2 = \frac{8 - 4\sqrt{6}}{-2} \][/tex]
Evaluating these roots numerically:
[tex]\[ x_1 \approx -8.9 \][/tex]
[tex]\[ x_2 \approx 0.9 \][/tex]
### Final Solution
Based on the calculations and the final answer provided:
- Vertex: [tex]\((-4, 24)\)[/tex]
- Roots: [tex]\((0.9, 0)\)[/tex] and [tex]\((-8.9, 0)\)[/tex]
Organizing the given values:
- Vertex: [tex]\((-4, 24)\)[/tex]
- Solutions (Roots): [tex]\((0.9, 0)\)[/tex] and [tex]\((-8.9, 0)\)[/tex]
Now let's fill in the solutions least to greatest (left to right).
Vertex: [tex]\((-4, 24)\)[/tex]
[tex]\(-8.9\)[/tex] 1)
[tex]\[
\text{Solutions: } (-8.9, 0) \text{ and } (0.9, 0)
\][/tex]
