To identify the general shape of the graph associated with the equation \( y = 3x^2 - 24x + 48 \), we can analyze the characteristics of the quadratic function.
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Coefficient of \(x^2\): The coefficient of \( x^2 \) is 3, which is positive. This indicates that the graph opens upward.
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Number of solutions: The equation has solutions \( x = 4 \) and \( x = 4 \), indicating that it has a double root. This means the vertex of the parabola lies at \( x = 4 \), and it touches the x-axis at this point but does not cross it.
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Evaluating the vertex: We can find the vertex by substituting \( x = 4 \) back into the equation to find the corresponding \( y \): \[ y = 3(4)^2 - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0. \] Therefore, the vertex is at the point \( (4, 0) \).
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Location of the vertex: Since the vertex is at \( (4, 0) \), it is to the right of the y-axis (x = 0).
Now we can summarize the options provided:
- The graph opens upward. (Correct)
- The graph opens downward. (Incorrect)
- The entire graph lies above the x-axis. (Incorrect, since it touches the x-axis at the vertex)
- The vertex is to the left of the y-axis. (Incorrect, since the vertex is at \( (4, 0) \) which is to the right of the y-axis)
Conclusion: The correct identification of the general shape of the associated graph is:
- The graph opens upward.
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