To analyze the quadratic equation \( y = x^2 - 8x \), we can determine various key points on the graph, such as x-intercepts, y-intercept, and the vertex.
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Vertex: The vertex of a quadratic equation \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
Here, \( a = 1 \) and \( b = -8 \): \[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \] To find the y-coordinate of the vertex, substitute \( x = 4 \) into the equation: \[ y = 4^2 - 8 \cdot 4 = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).
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Y-intercept: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 8 \cdot 0 = 0 \] So the y-intercept is \( (0, 0) \).
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X-intercepts: The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 8x \] Factoring gives: \[ 0 = x(x - 8) \] Thus, the solutions are \( x = 0 \) and \( x = 8 \), resulting in x-intercepts at \( (0, 0) \) and \( (8, 0) \).
In summary, the key points for the graph of the equation \( y = x^2 - 8x \) are:
- Vertex: \( (4, -16) \)
- X-intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-intercept: \( (0, 0) \)
Among the provided responses, the correct one captures these points accurately:
- x-intercepts: \( (0,0) \) and \( (8,0) \)
- y-intercept: \( (0,0) \)
- vertex: \( (4,-16) \)
1 answer