To analyze the given quadratic function \(y = x^2 - 8x\), we need to determine the vertex, x-intercepts, and y-intercept.
1. Vertex
The vertex form of a quadratic equation \(y = ax^2 + bx + c\) can be analyzed using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] For our equation \(y = x^2 - 8x\), the coefficients are \(a = 1\) and \(b = -8\).
Calculating the x-coordinate of the vertex: \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] Now substitute \(x = 4\) back into the original equation to find the y-coordinate: \[ y = (4)^2 - 8 \cdot 4 = 16 - 32 = -16 \] Thus, the vertex is at: \[ \text{Vertex: } (4, -16) \]
2. X-Intercepts
To find the x-intercepts, set \(y = 0\): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] This gives the solutions: \[ x = 0 \quad \text{or} \quad x = 8 \] Thus, the x-intercepts are: \[ \text{X-Intercepts: } (0, 0), (8, 0) \]
3. Y-Intercept
To find the y-intercept, set \(x = 0\): \[ y = (0)^2 - 8 \cdot 0 = 0 \] Thus, the y-intercept is: \[ \text{Y-Intercept: } (0, 0) \]
Summary of Key Points
- Vertex: (4, -16)
- X-Intercepts: (0, 0) and (8, 0)
- Y-Intercept: (0, 0)
Thus, from the provided response options, the correct response would be:
- x-intercepts: (0, 0), (8, 0)
- y-intercept: (0, 0)
- vertex: (4, -16)
1 answer