$\frac1{x^2}$ 永远为正，而且左右对称。减去 $9$ 表示整张图像向下平移 $9$ 个单位。因为 $x=0$ 时无定义，所以 $x=0$ 是竖直渐近线；当 $|x|$ 很大时，$\frac1{x^2}\to0$，所以 $y=-9$ 是水平渐近线。

For the $x$-intercepts, set $y=0$:

So the $x$-intercepts are

There is no $y$-intercept because $x=0$ is not in the domain.

The asymptotes are

The sketch should have two branches, symmetric about the $y$-axis, approaching $x=0$ vertically and approaching $y=-9$ horizontally.

交点满足两条曲线的 $y$ 相等。整理后会得到关于 $x$ 的四次方程。因为里面只出现 $x^2$ 和 $x^4$，可以令 $u=x^2$，这样变成关于 $u$ 的二次方程。

要有 $4$ 个不同交点，关于 $u$ 的方程需要有两个不同的正根；每个正的 $u$ 都会给出 $x=\pm\sqrt u$ 两个不同的 $x$ 值。

At the points of intersection,

Multiply by $x^2$:

Let

Then

For the curve $y=kx^2$ to meet $C$ at 4 distinct points, the equation in $u$ must have two distinct positive roots.

First, $k$ must be negative. This makes the product of the two roots

positive, and the sum of the roots

positive.

The roots must also be distinct, so the discriminant must be positive:

Combining this with $k<0$ gives