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 touches the x-axis exactly twice. The graph touches the x -axis exactly twice. The graph opens downward. The graph opens downward. The graph lies above the x-axis.

Answer 1

To analyze the graph of the quadratic equation given, \( y = x^2 - 16x + 89 \), we can explore several key features:

1. **Finding the vertex**: The vertex of a quadratic equation in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( a = 1 \) and \( b = -16 \).
- Thus, \( x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \).
- The vertex's x-coordinate is 8, which is to the right of the y-axis.

2. **Zeros of the function**: The given zeros are \( x = 8 - 5i \) and \( x = 8 + 5i \). Since these are complex numbers, it means the quadratic does not intersect the x-axis and the graph does not touch or cross the x-axis at any real points.

3. **Opening direction**: The coefficient of \( x^2 \) is positive (\( a = 1 \)), so the parabola opens upwards.

4. **Position relative to the x-axis**: Since the graph does not touch the x-axis and the vertex is at \( (8, y) \) for some positive \( y \) (since the parabola opens upwards and does not have real zeros), the graph lies entirely above the x-axis.

To summarize:

- The vertex is located at \( x = 8 \) (to the right of the y-axis).
- The graph does not touch the x-axis (it has complex zeros).
- The graph opens upwards and lies entirely above the x-axis.

Given these observations, the correct identification from the provided responses is:

**The graph lies above the x-axis**.