To find the key points on the graph of the quadratic function \( y = x^2 - 8x \), let's identify the vertex, x-intercepts, and y-intercept step-by-step.
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Vertex: The vertex of a parabola given in the form \( y = ax^2 + bx + c \) 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 \( x = 4 \) back into the equation: \[ y = (4)^2 - 8(4) = 16 - 32 = -16 \] Thus, the vertex is \( (4, -16) \).
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Y-Intercept: The y-intercept occurs when \( x = 0 \): \[ y = (0)^2 - 8(0) = 0 \] 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 the equation: \[ 0 = x(x - 8) \] This gives us \( x = 0 \) and \( x = 8 \). Therefore, the x-intercepts are \( (0, 0) \) and \( (8, 0) \).
Summary of the key points:
- Vertex: \( (4, -16) \)
- X-Intercepts: \( (0, 0) \) and \( (8, 0) \)
- Y-Intercept: \( (0, 0) \)
Now, let's evaluate the provided options based on this information:
- Vertex: (4, -16) (correct)
- X-Intercepts: (0, 0) and (8, 0) (correct)
- Y-Intercept: (0, 0) (correct)
Thus, the accurate response is:
- X-intercepts: (0, 0) and (8, 0)
- Vertex: (4, -16)
- Y-intercept: (0, 0)