Answer :
To find the vertex and [tex]\( x \)[/tex]-intercepts of the quadratic function [tex]\( y = x^2 - 6x - 7 \)[/tex], let's go through the detailed steps:
### Finding the Vertex
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex].
Calculating [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 3 \)[/tex] into the equation [tex]\( y = x^2 - 6x - 7 \)[/tex]:
[tex]\[ y = (3)^2 - 6(3) - 7 = 9 - 18 - 7 = -16 \][/tex]
Thus, the vertex is:
[tex]\[ (3, -16) \][/tex]
### Finding the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are the points where the graph crosses the [tex]\( x \)[/tex]-axis, which occurs when [tex]\( y = 0 \)[/tex]. To find these points, we need to solve the quadratic equation [tex]\( x^2 - 6x - 7 = 0 \)[/tex].
We use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -7 \)[/tex].
First, calculate the discriminant:
[tex]\[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-7) = 36 + 28 = 64 \][/tex]
Then, calculate the two solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6 \pm \sqrt{64}}{2 \cdot 1} = \frac{6 \pm 8}{2} \][/tex]
This gives us:
[tex]\[ x = \frac{6 + 8}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{6 - 8}{2} = \frac{-2}{2} = -1 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ (7, 0) \text{ and } (-1, 0) \][/tex]
### Summary
Based on the calculations, the solutions are:
- Vertex: [tex]\((3, -16)\)[/tex]
- [tex]\( x \)[/tex]-intercepts: [tex]\((7, 0) \text{ and } (-1, 0)\)[/tex]
Therefore, the correct answers are:
- B. Vertex: [tex]\((3, -16)\)[/tex]
- A. [tex]\( x \)[/tex]-intercepts: [tex]\(((-1,0), (7,0))\)[/tex]
### Finding the Vertex
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex].
Calculating [tex]\( x \)[/tex]:
[tex]\[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 3 \)[/tex] into the equation [tex]\( y = x^2 - 6x - 7 \)[/tex]:
[tex]\[ y = (3)^2 - 6(3) - 7 = 9 - 18 - 7 = -16 \][/tex]
Thus, the vertex is:
[tex]\[ (3, -16) \][/tex]
### Finding the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are the points where the graph crosses the [tex]\( x \)[/tex]-axis, which occurs when [tex]\( y = 0 \)[/tex]. To find these points, we need to solve the quadratic equation [tex]\( x^2 - 6x - 7 = 0 \)[/tex].
We use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -7 \)[/tex].
First, calculate the discriminant:
[tex]\[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot (-7) = 36 + 28 = 64 \][/tex]
Then, calculate the two solutions for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6 \pm \sqrt{64}}{2 \cdot 1} = \frac{6 \pm 8}{2} \][/tex]
This gives us:
[tex]\[ x = \frac{6 + 8}{2} = \frac{14}{2} = 7 \][/tex]
[tex]\[ x = \frac{6 - 8}{2} = \frac{-2}{2} = -1 \][/tex]
So, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ (7, 0) \text{ and } (-1, 0) \][/tex]
### Summary
Based on the calculations, the solutions are:
- Vertex: [tex]\((3, -16)\)[/tex]
- [tex]\( x \)[/tex]-intercepts: [tex]\((7, 0) \text{ and } (-1, 0)\)[/tex]
Therefore, the correct answers are:
- B. Vertex: [tex]\((3, -16)\)[/tex]
- A. [tex]\( x \)[/tex]-intercepts: [tex]\(((-1,0), (7,0))\)[/tex]
