To determine the general shape of the graph of the quadratic equation \( y = 3x^2 - 24x + 48 \), we need to analyze its properties.
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Coefficient of \( x^2 \): The equation is of the form \( y = ax^2 + bx + c \), where \( a = 3 \), \( b = -24 \), and \( c = 48 \). Since \( a > 0 \), the graph opens upward.
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Vertex Location: The x-coordinate of the vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( -\frac{b}{2a} = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4 \).
- Since the vertex's x-coordinate (which is 4) is on the right side of the y-axis, we can say that the vertex is to the right of the y-axis.
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Solutions: The given solutions \( x = 4 \) are repeated (indicating that the vertex is located at that point in terms of x, which means the graph touches the x-axis at \( (4, 0) \) and does not cross it).
Therefore, we can summarize the results:
- The graph opens upward.
- The vertex is located on the x-axis and not to the left of the y-axis.
- The graph does cross the x-axis at \( (4, 0) \), meaning that it does not lie entirely above the x-axis.
From the options provided, the appropriate response is "The graph opens upward."
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