Answer :
To analyze the given quadratic function [tex]\( f(x) = -x^2 + 2x - 8 \)[/tex], let's identify its key features step-by-step:
1. Vertex: The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex]. For the given function, the coefficients are:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = -8 \)[/tex]
Therefore, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{2}{2(-1)} = 1 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] into the quadratic function:
[tex]\[ f(1) = -1^2 + 2(1) - 8 = -1 + 2 - 8 = -7 \][/tex]
Thus, the vertex is [tex]\((1, -7)\)[/tex].
2. Roots: The roots (or zeros) of a quadratic function are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. To find these roots, solve the equation [tex]\( -x^2 + 2x - 8 = 0 \)[/tex].
This can be expressed as:
[tex]\[ x = \frac{2 \pm \sqrt{2^2 - 4(-1)(-8)}}{2(-1)} \][/tex]
Simplifying inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{4 - 32}}{-2} = \frac{2 \pm \sqrt{-28}}{-2} \][/tex]
The roots are complex numbers:
[tex]\[ x = \frac{2 \pm \sqrt{-28}}{-2} = \frac{2 \pm 2i\sqrt{7}}{-2} = -1 \pm i\sqrt{7} \][/tex]
So, the roots are [tex]\( 1 - \sqrt{7}i \)[/tex] and [tex]\( 1 + \sqrt{7}i \)[/tex].
3. Direction of Opening: Since the coefficient of [tex]\( x^2 \)[/tex] (i.e., [tex]\( a \)[/tex]) is negative ([tex]\( a = -1 \)[/tex]), the parabola opens downwards.
4. Y-intercept: The y-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. Simply substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -(0)^2 + 2(0) - 8 = -8 \][/tex]
To summarize:
- The vertex is at [tex]\((1, -7)\)[/tex].
- The roots are [tex]\( 1 - \sqrt{7}i \)[/tex] and [tex]\( 1 + \sqrt{7}i \)[/tex].
- The parabola opens downwards.
- The y-intercept is [tex]\(-8\)[/tex].
You can now accurately choose the correct answers for each key feature from the drop-down menus!
1. Vertex: The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex]. For the given function, the coefficients are:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = -8 \)[/tex]
Therefore, the x-coordinate of the vertex is:
[tex]\[ x = -\frac{2}{2(-1)} = 1 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 1 \)[/tex] into the quadratic function:
[tex]\[ f(1) = -1^2 + 2(1) - 8 = -1 + 2 - 8 = -7 \][/tex]
Thus, the vertex is [tex]\((1, -7)\)[/tex].
2. Roots: The roots (or zeros) of a quadratic function are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. To find these roots, solve the equation [tex]\( -x^2 + 2x - 8 = 0 \)[/tex].
This can be expressed as:
[tex]\[ x = \frac{2 \pm \sqrt{2^2 - 4(-1)(-8)}}{2(-1)} \][/tex]
Simplifying inside the square root:
[tex]\[ x = \frac{2 \pm \sqrt{4 - 32}}{-2} = \frac{2 \pm \sqrt{-28}}{-2} \][/tex]
The roots are complex numbers:
[tex]\[ x = \frac{2 \pm \sqrt{-28}}{-2} = \frac{2 \pm 2i\sqrt{7}}{-2} = -1 \pm i\sqrt{7} \][/tex]
So, the roots are [tex]\( 1 - \sqrt{7}i \)[/tex] and [tex]\( 1 + \sqrt{7}i \)[/tex].
3. Direction of Opening: Since the coefficient of [tex]\( x^2 \)[/tex] (i.e., [tex]\( a \)[/tex]) is negative ([tex]\( a = -1 \)[/tex]), the parabola opens downwards.
4. Y-intercept: The y-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. Simply substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -(0)^2 + 2(0) - 8 = -8 \][/tex]
To summarize:
- The vertex is at [tex]\((1, -7)\)[/tex].
- The roots are [tex]\( 1 - \sqrt{7}i \)[/tex] and [tex]\( 1 + \sqrt{7}i \)[/tex].
- The parabola opens downwards.
- The y-intercept is [tex]\(-8\)[/tex].
You can now accurately choose the correct answers for each key feature from the drop-down menus!
