The equation given is a quadratic equation in the form \(y = ax^2 + bx + c\), where \(a = 3\), \(b = -24\), and \(c = 48\).
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Determine if the graph opens upward or downward: Since the coefficient of \(x^2\) (which is \(a\)) is positive (\(a = 3\)), the graph opens upward.
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Vertex position: The vertex of a quadratic function \(y = ax^2 + bx + c\) can be found using the formula: \[ x_{vertex} = -\frac{b}{2a} \] Substituting the values of \(a\) and \(b\): \[ x_{vertex} = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4 \] This means the vertex is located at \(x = 4\), which is on the right side of the y-axis.
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Position of the entire graph relative to the x-axis: To determine if the entire graph lies above the x-axis, we can assess the y-values at the solutions (roots) \(x = 4\): \[ y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0 \] Since the vertex represents the minimum point of the graph, and the graph opens upwards, with the minimum value (the vertex) being at \(y = 0\) when \(x = 4\), it means that the graph touches the x-axis at \(y = 0\), but does not lie entirely above the x-axis — it intersects the x-axis.
Based on the above analysis, the correct identification of the general shape of the graph is:
- The graph opens upward.