The graph of the parabola $y=x^{2}-5 x+9$ is rotated $180^{\circ}$ about the origin. What is the equation of the new parabola?
(a) $y=x^{2}+5 x+9$
(c) $y=-x^{2}+5 x-9$
(e) $y=-x^{2}-5 x-9$
(b) $y=x^{2}-5 x-9$
(d) $y=-x^{2}-5 x+9$
The correct option is (e).

A rotation of $180^{\circ}$ can be visualized as a half-turn. Here we have a half-turn around the origin. The figure illustrates what a half-turn does to the coordinates of points in the plane. For example, the point $A^{\prime}$ is the result of the half-turn applied to the point $A$; in other words, $A^{\prime}$ is where the point $A$ ends up after the half-turn. Similarly, $B^{\prime}$ is where $B$ will end up after the half-turn.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-050.jpg?height=520&width=674&top_left_y=1810&top_left_x=1159)

It is easy to see that in the transition from $A$ to $A^{\prime}$, the coordinates change sign. Thus, we see that a half-turn around the origin takes any point $(x, y)$ to the point $(-x, -y)$. Therefore, $(a, b)$ belongs to the new parabola if and only if $(-a, -b)$ belongs to the original parabola $y = x^2 - 5x + 9$, which means $-b = a^2 + 5a + 9$ or, equivalently, $b = -a^2 - 5a - 9$. Hence, the equation of the new parabola is $y = -x^2 - 5x - 9$.