Bot GPT 4-o mini The equation given is \( y = 3x^2 - 24x + 48 \).
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Determine the direction the graph opens:
- The coefficient of \( x^2 \) is positive (3), which means the graph opens upward.
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Identify the vertex:
- The vertex can be calculated using the formula \( x = -\frac{b}{2a} \) where \( a = 3 \) and \( b = -24 \): \[ x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4. \] The vertex is at (4, y-value), which means it is to the right of the y-axis.
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Analyze the y-value at the vertex:
- Substitute \( x = 4 \) back into the equation to find the y-value at the vertex: \[ y = 3(4)^2 - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0. \] The vertex is at (4, 0), which means the graph touches the x-axis.
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Conclude the overall shape of the graph:
- Since the graph opens upward and the vertex is on the x-axis, the graph lies above the x-axis except at the vertex where it touches it.
Given this information, the most accurate statement from the provided options is:
The graph opens upward.
1 answer