Asked by yeslian
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 touches the x-axis exactly twice.
The graph touches the x -axis exactly twice.
The graph lies above the x-axis.
The graph lies above the x -axis.
The graph opens downward.
Responses
The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.
The graph touches the x-axis exactly twice.
The graph touches the x -axis exactly twice.
The graph lies above the x-axis.
The graph lies above the x -axis.
The graph opens downward.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the given equation and determine the general shape of its associated graph, let's break down the information provided.
1. **Identifying the zeros**: The equation \( y = x^2 - 16x + 89 \) has complex zeros at \( x = 8 - 5i \) and \( x = 8 + 5i \). Since the roots are complex, this means that the graph does not intersect the x-axis at any point.
2. **Shape of the graph**: Since the leading coefficient of \( x^2 \) (which is 1) is positive, the graph is a parabola that opens upward.
3. **Vertex**: The vertex of the parabola can be determined from the standard form of the quadratic equation \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). For the given equation, \( a = 1 \) and \( b = -16 \).
\[
x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Since the x-coordinate of the vertex is 8, which is to the right of the y-axis, the vertex is indeed to the right of the y-axis.
4. **Position of the graph**: Given that the graph does not cross the x-axis (due to complex roots), it remains entirely above the x-axis.
Based on this analysis, the correct response to identify the general shape of the graph is:
- **The graph lies above the x-axis.**
1. **Identifying the zeros**: The equation \( y = x^2 - 16x + 89 \) has complex zeros at \( x = 8 - 5i \) and \( x = 8 + 5i \). Since the roots are complex, this means that the graph does not intersect the x-axis at any point.
2. **Shape of the graph**: Since the leading coefficient of \( x^2 \) (which is 1) is positive, the graph is a parabola that opens upward.
3. **Vertex**: The vertex of the parabola can be determined from the standard form of the quadratic equation \( y = ax^2 + bx + c \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). For the given equation, \( a = 1 \) and \( b = -16 \).
\[
x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Since the x-coordinate of the vertex is 8, which is to the right of the y-axis, the vertex is indeed to the right of the y-axis.
4. **Position of the graph**: Given that the graph does not cross the x-axis (due to complex roots), it remains entirely above the x-axis.
Based on this analysis, the correct response to identify the general shape of the graph is:
- **The graph lies above the x-axis.**
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