Question
Given the equation y=x2−16x+89 with zeros of x=8−5i and x=8+5i , which of the following identifies the general shape of its associated graph?(1 point)
Responses
The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.
The graph lies above the x-axis.
The graph lies above the x -axis.
The graph opens downward.
The graph opens downward.
The graph touches the x-axis exactly twice.
Responses
The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.
The graph lies above the x-axis.
The graph lies above the x -axis.
The graph opens downward.
The graph opens downward.
The graph touches the x-axis exactly twice.
Answers
Answered by
GPT-4o mini
To analyze the general shape of the graph associated with the given equation \( y = x^2 - 16x + 89 \) and its zeros \( x = 8 - 5i \) and \( x = 8 + 5i \), we can consider the following points:
1. **Complex Zeros**: Since the zeros are complex (with imaginary parts), it indicates that the graph does not intersect the x-axis. Hence, the graph does not touch the x-axis at all.
2. **Direction of the Parabola**: The coefficient of \( x^2 \) is positive (1 in this case), which means the parabola opens upwards.
3. **Vertex**: To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -16 \):
\[
x_{\text{vertex}} = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Since the vertex is at \( x = 8 \), it is on the right side of the y-axis.
Given these points, we can summarize:
- The graph does not touch the x-axis (or intersect it).
- The graph opens upward, not downward.
- The vertex is not left of the y-axis; in fact, it’s right at \( x = 8 \).
Based on this analysis, the correct response that identifies the general shape of the graph is:
**The graph lies above the x-axis.**
1. **Complex Zeros**: Since the zeros are complex (with imaginary parts), it indicates that the graph does not intersect the x-axis. Hence, the graph does not touch the x-axis at all.
2. **Direction of the Parabola**: The coefficient of \( x^2 \) is positive (1 in this case), which means the parabola opens upwards.
3. **Vertex**: To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -16 \):
\[
x_{\text{vertex}} = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Since the vertex is at \( x = 8 \), it is on the right side of the y-axis.
Given these points, we can summarize:
- The graph does not touch the x-axis (or intersect it).
- The graph opens upward, not downward.
- The vertex is not left of the y-axis; in fact, it’s right at \( x = 8 \).
Based on this analysis, the correct response that identifies the general shape of the graph is:
**The graph lies above the x-axis.**
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