To determine the general shape of the graph based on the given quadratic equation \( y = 3x^2 - 24x + 48 \), we can analyze the equation:
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Determine the orientation of the graph: The coefficient of \( x^2 \) is \( 3 \), which is positive. This means the graph opens upward.
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Identify the vertex: Since the quadratic can be expressed in vertex form or by finding the vertex using the formula \( x = -\frac{b}{2a} \), where \( a = 3 \) and \( b = -24 \): \[ x = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4 \] The vertex's x-coordinate is \( 4 \), meaning it is not to the left of the y-axis (it is actually on the right side since \( 4 > 0 \)).
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Check if the graph lies above the x-axis: To assess if the graph lies above the x-axis, we can find the y-value of the vertex: \[ y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0 \] The vertex is at \( (4, 0) \), indicating that it touches the x-axis at this point. Therefore, the graph does not lie entirely above the x-axis.
Given this analysis, the only statement that accurately describes the graph based on the information is:
The graph opens upward.
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