To analyze the quadratic function \( y = x^2 - 8x \) and find the key points, we can start by rewriting the equation into vertex form or finding the relevant intercepts directly.
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Find the vertex: The vertex form of a parabola can be derived from the standard form \( y = ax^2 + bx + c \). The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = -8 \). \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] Now plug \( x = 4 \) back into the original equation to find the y-coordinate: \[ y = (4)^2 - 8 \cdot 4 = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).
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Find the x-intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 8x \] Factor the equation: \[ 0 = x(x - 8) \] This gives x-intercepts at \( x = 0 \) and \( x = 8 \), resulting in the points: \[ (0, 0) \quad \text{and} \quad (8, 0) \]
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Find the y-intercept: To find the y-intercept, set \( x = 0 \): \[ y = 0^2 - 8 \cdot 0 = 0 \] Thus, the y-intercept is \( (0, 0) \).
Summary of Key Points:
- Vertex: \( (4, -16) \)
- x-intercepts: \( (0, 0) \) and \( (8, 0) \)
- y-intercept: \( (0, 0) \)
Final Response:
The correct option is:
- x-intercepts: \( (0, 0), (8, 0) \)
- y-intercept: \( (0, 0) \)
- vertex: \( (4, -16) \)
1 answer