The equation you provided is a quadratic function of the form \(y = ax^2 + bx + c\). In your case, the equation is:
\[ y = x^2 - 16x + 89 \]
Given that the zeros of the function are complex (specifically \(x = 8 - 5i\) and \(x = 8 + 5i\)), this indicates that the graph of the function does not intersect the x-axis. Instead, since the zeros are complex and have no real parts, the graph opens upward because the coefficient of \(x^2\) (which is 1) is positive.
Now let's consider the responses provided:
- The graph touches the x-axis exactly twice. (Incorrect)
- The graph opens downward. (Incorrect)
- The vertex is to the left of the y-axis. (Potentially correct, but we will address in a bit)
Since the graph opens upward and does not touch the x-axis, the statements about touching the x-axis are incorrect.
To determine the position of the vertex, we can find the x-coordinate of the vertex using the formula \(x = -\frac{b}{2a}\):
Here, \(a = 1\) and \(b = -16\):
\[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \]
The vertex is at \(x = 8\), which is on the y-axis, not to the left.
Conclusion:
- The graph does not touch the x-axis at all.
- The graph opens upward.
- The vertex is on the y-axis.
Thus, it seems the most accurate response among the choices you gave would be that the graph opens upward, though that wasn't one of the specified choices. From what you provided, no option accurately describes the graph.