To analyze the graph of the quadratic equation \( y = x^2 - 16x + 89 \), we can start by determining the general characteristics of the graph based on its roots (or zeros) and the equation itself.
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Roots: The equation has complex roots \( x = 8 - 5i \) and \( x = 8 + 5i \). Since these roots are complex (not real), it indicates that the graph does not intersect the x-axis at any point. Therefore, the graph must lie entirely above or entirely below the x-axis.
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Quadratic Coefficient: The coefficient of \( x^2 \) is positive (1 in this case). This means the graph opens upwards.
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Vertex: The vertex of a quadratic equation in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -16 \): \[ x = -\frac{-16}{2 \times 1} = 8 \] The vertex \( x \)-coordinate is 8, which is to the right of the y-axis.
Given this analysis, we can make the following conclusions:
- The graph does not touch or intersect the x-axis (since it has complex roots).
- Therefore, the graph does not touch the x-axis exactly twice.
- The vertex is at x = 8, which is to the right of the y-axis.
- Since the parabola opens upward and there are no x-intercepts, the graph lies entirely above the x-axis.
Based on the responses provided, the most applicable conclusion is:
The graph lies above the x-axis.
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