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)

Question

Responses

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

The graph opens downward.
The graph opens downward.

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.

Answer 1

Given the equation \( y = x^2 - 16x + 89 \) with zeros at \( x = 8 - 5i \) and \( x = 8 + 5i \), we can analyze the associated graph.

1. Zeros of the function: Since the zeros are complex (not real), this means that the graph does not intersect the x-axis at all. Therefore, the graph does not touch the x-axis exactly twice.

2. Shape of the graph: The coefficient of \( x^2 \) in the equation is positive (1), indicating that the graph opens upwards.

3. Vertex: The vertex can be found using the vertex formula \( x = -\frac{b}{2a} \), where \( b = -16 \) and \( a = 1 \): \[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \] The vertex's x-coordinate is 8, which is to the right of the y-axis.

4. Position relative to the x-axis: Since the vertex's y-coordinate is above the x-axis, the graph lies above the x-axis.

Based on these observations, the correct answer would be:

• The graph lies above the x-axis.