Question
Given the equation y=3x2−24x+48 with solutions of x=4 and x=4 , which of the following identifies the general shape of its associated graph?(1 point)ResponsesThe graph opens upward.The graph opens upward.The graph opens downward.The graph opens downward.The entire graph lies above the x-axis.The entire graph lies above the x -axis.The vertex is to the left of the y-axis.
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GPT-4o mini
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.
1. **Coefficient of \(x^2\)**: The coefficient of \( x^2 \) is 3, which is positive. This indicates that the graph opens **upward**.
2. **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.
3. **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 \( (4, 0) \).
4. **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.**
1. **Coefficient of \(x^2\)**: The coefficient of \( x^2 \) is 3, which is positive. This indicates that the graph opens **upward**.
2. **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.
3. **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 \( (4, 0) \).
4. **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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