Question

the equation $y = -2x^2 + 8x - 12$ has solutions $x = 2 + i$ and $x = 2 - i$. what is the general shape of its graph?
(1 point)

an upward - opening parabola that does not cross the x - axis.
a downward - opening parabola that crosses the x - axis twice.
an upward - opening parabola that touches the x - axis once.
a downward - opening parabola that does not cross the x - axis.

Explanation:

Step1: Determine the direction of the parabola

For a quadratic equation in the form \( y = ax^2+bx + c \), the coefficient \( a \) determines the direction the parabola opens. If \( a>0 \), it opens upward; if \( a < 0 \), it opens downward. In the equation \( y=-2x^{2}+8x - 12 \), \( a=-2 \), which is less than 0. So the parabola opens downward.

Step2: Analyze the x - axis intersection

The solutions of the quadratic equation (when \( y = 0 \)) are given as \( x = 2 + i \) and \( x=2 - i \), which are complex numbers (since they involve the imaginary unit \( i \)). This means that the graph of the quadratic function does not intersect the real x - axis (because the roots are not real).

Answer:

A downward - opening parabola that does not cross the x - axis. (The last option: A downward - opening parabola that does not cross the x - axis)