To analyze the quadratic function \( y = x^2 - 8x \), we first need to rewrite it in vertex form by completing the square, or we can simply utilize the formula for the vertex and the intercepts.
1. Finding the Vertex
The standard form of a quadratic equation is given by: \[ y = ax^2 + bx + c \] where \( a = 1 \), \( b = -8 \), and \( c = 0 \).
The x-coordinate of the vertex can be found using: \[ x = -\frac{b}{2a} = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \]
Substituting \( x = 4 \) back into the equation to get the y-coordinate: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).
2. Finding the x-intercepts
To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] This yields two solutions: \[ x = 0 \quad \text{and} \quad x = 8 \] Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
3. Finding the y-intercept
To find the y-intercept, set \( x = 0 \): \[ y = (0)^2 - 8(0) = 0 \] So, the y-intercept is \( (0, 0) \).
Summary
- Vertex: \( (4, -16) \)
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
Selecting the Correct Response
From the available options, the correct response would be:
- Vertex: \( (4, -16) \)
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
There's no mention of the alternative combinations listed in your choices; however, the key points are definitively represented here.