Sure, let’s break down the quadratic function [tex]\( f(x) = -x^2 - 2x + 8 \)[/tex] step-by-step to find the required intercepts, vertex, and line of symmetry.
### Step 1: Finding the x-intercepts
To find the x-intercepts, we solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].
[tex]\[
-x^2 - 2x + 8 = 0
\][/tex]
This is a quadratic equation, and we can solve it using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 8 \)[/tex].
First, we calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-2)^2 - 4(-1)(8) = 4 + 32 = 36
\][/tex]
Using the quadratic formula, we get:
[tex]\[
x = \frac{-(-2) \pm \sqrt{36}}{2(-1)} = \frac{2 \pm 6}{-2}
\][/tex]
This results in two solutions:
[tex]\[
x_1 = \frac{2 + 6}{-2} = \frac{8}{-2} = -4
\][/tex]
[tex]\[
x_2 = \frac{2 - 6}{-2} = \frac{-4}{-2} = 2
\][/tex]
The smallest (leftmost) [tex]\( x \)[/tex]-intercept is [tex]\( x = -4 \)[/tex].
The largest (rightmost) [tex]\( x \)[/tex]-intercept is [tex]\( x = 2 \)[/tex].
### Step 2: Finding the y-intercept
The y-intercept is found by evaluating [tex]\( f(0) \)[/tex]:
[tex]\[
f(0) = -0^2 - 2 \cdot 0 + 8 = 8
\][/tex]
The [tex]\( y \)[/tex]-intercept is [tex]\( y = 8 \)[/tex].
### Step 3: Finding the vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the vertex [tex]\( x \)[/tex]-coordinate, [tex]\( x = -\frac{b}{2a} \)[/tex].
For [tex]\( a = -1 \)[/tex] and [tex]\( b = -2 \)[/tex]:
[tex]\[
x = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1
\][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, we substitute [tex]\( x = -1 \)[/tex] back into the function:
[tex]\[
f(-1) = -(-1)^2 - 2(-1) + 8 = -1 + 2 + 8 = 9
\][/tex]
The vertex is [tex]\((-1, 9)\)[/tex].
### Step 4: Finding the line of symmetry
The line of symmetry for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is the vertical line that passes through the vertex, given by [tex]\( x = -\frac{b}{2a} \)[/tex].
Hence, the line of symmetry is:
[tex]\[
x = -1
\][/tex]
Summarizing all the results:
- The smallest (leftmost) [tex]\( x \)[/tex]-intercept is [tex]\( x = -4 \)[/tex].
- The largest (rightmost) [tex]\( x \)[/tex]-intercept is [tex]\( x = 2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept is [tex]\( y = 8 \)[/tex].
- The vertex is [tex]\( (-1, 9) \)[/tex].
- The line of symmetry has the equation [tex]\( x = -1 \)[/tex].
These are the required details for the quadratic function [tex]\( f(x) = -x^2 - 2x + 8 \)[/tex].
