To determine the general shape of the graph associated with the quadratic equation \( y = 3x^2 - 24x + 48 \), we can analyze the coefficient of the \( x^2 \) term and the properties of the quadratic.
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Direction of the graph: The leading coefficient (the coefficient of \( x^2 \)) is 3, which is positive. This means the graph opens upward.
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Vertex and X-Intercepts: The solutions (roots) of the equation are \( x = 4 \) and \( x = 4\) (which indicates a repeated root). This means the vertex also occurs at \( x = 4 \).
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Value at the vertex: To find the vertex, we can substitute \( x = 4 \) into the equation: \[ y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0. \] Thus, the vertex, which is also the minimum point of the graph because it opens upward, is at the point (4, 0). This means at the vertex, the graph touches the x-axis.
From this analysis, we can conclude:
- The graph opens upward.
- The vertex is at ground level (y=0), so part of the graph touches the x-axis, meaning it doesn't lie entirely above it.
Thus, the best choice from your options is:
The graph opens upward.
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