To determine the general shape of the graph for the equation \( y = 3x^2 - 24x + 48 \), we can analyze the quadratic equation.
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Determine the vertex and direction of opening: The standard form of a quadratic equation is \( y = ax^2 + bx + c \). In this case, \( a = 3 \). Since \( a > 0 \), the graph opens upward.
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Find the vertex: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = -24 \) and \( a = 3 \).
- \( x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4 \).
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Evaluate the vertex position: The vertex occurs at \( x = 4 \), which is to the right of the y-axis.
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Roots of the equation: The given solutions (or roots) are \( x = 4 \) and \( x = 4 \), meaning the graph touches the x-axis at one point (the vertex) but does not go below it.
With this analysis, the graph:
- Opens upward.
- The entire graph is above the x-axis except for the vertex point.
Given the options:
- The graph opens upward.
Thus, the correct response is: The graph opens upward.