这是 reciprocal graph。分母为 $0$ 的位置给出竖直渐近线 $x=k$。由于 $k>0$，这条渐近线在 $y$ 轴右侧。令 $x=0$ 可得 $y$ 轴截距；但分子恒为 $4$，所以不会有 $x$ 轴截距。

The vertical asymptote occurs when the denominator is zero:

The $y$-intercept is found by setting $x=0$:

So the curve cuts the $y$-axis at

There is no $x$-intercept, since

has no solution.

The sketch should show a decreasing reciprocal curve with vertical asymptote $x=k$, a right-hand branch above the $x$-axis, and a left-hand branch passing through $\left(0,-\frac4k\right)$.

直线不与曲线相交或相切，意思是联立后没有实根。联立得到关于 $x$ 的二次方程，再令判别式小于 $0$。

At an intersection,

Multiplying by $x-k$,

Hence

For the line not to cross or touch the curve, this quadratic must have no real roots, so

Therefore