曲线 $C$ 的方程为 $y=\dfrac{10x^2-11x-18}{10x-18}$。

(a) 求 $C$ 的渐近线方程。

(b) 证明 $C$ 没有驻点。

(c) 画出 $C$ 的草图，并写出曲线与坐标轴交点的坐标。

(d) 画出方程为 $y=\Big|\dfrac{10x^2-11x-18}{10x-18}\Big|$ 的曲线的草图。

(e) 已知当 $\dfrac{-29-\sqrt{4441}}{20}<x<p$ 和 $\dfrac{-29+\sqrt{4441}}{20}<x<q$ 时，$\Big|\dfrac{10x^2-11x-18}{10x-18}\Big|<4$，求 $p$ 和 $q$ 的值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

分母零点与分子不能约去。多项式除法给出斜渐近线，余项要写清楚。

Polynomial division gives

At $x=9/5$, the numerator is $-27/5\ne0$, so the vertical asymptote is $\boxed{x=9/5}$.

The remainder term tends to zero as $|x|\to\infty$, giving the oblique asymptote $\boxed{y=x+7/10}$.

对拆分后的函数求导，导数可直接写成1加正数，因此不可能为零。

Differentiating the expression in (a),

for every $x\ne9/5$. Thus the derivative never vanishes on the domain, and $\boxed{C\text{ has no stationary points}}$.

先因式分解分子求横轴交点，再用渐近线与严格递增性画两支。

The numerator factorises as

Hence the intercepts are $\boxed{(-9/10,0)}$, $\boxed{(2,0)}$ and $\boxed{(0,1)}$.

Sketch two increasing branches with asymptotes $x=9/5$ and $y=x+7/10$. The left branch lies above the oblique asymptote, crosses the axes at $(-9/10,0)$ and $(0,1)$, and tends to $+\infty$ as $x\to(9/5)^-$. The right branch lies below the oblique asymptote, starts from $-\infty$ as $x\to(9/5)^+$, crosses $(2,0)$ and increases towards the oblique asymptote.

反射函数值为负的两段，两个横轴交点都变成尖点，竖直渐近线仍要保留。

Reflect the parts with $x<-9/10$ and $9/5<x<2$ in the $x$-axis; keep the positive parts unchanged.

The graph lies on or above the $x$-axis and has sharp minima at $(-9/10,0)$ and $(2,0)$. It passes through $(0,1)$, tends to $+\infty$ on both sides of $x=9/5$ and retains that vertical asymptote.

题目已给对应y=−4的两个边界，另外两个边界由y=4求出。用两支严格递增确认p、q的顺序。

The two stated lower endpoints come from $y=-4$. The upper endpoints satisfy $y=4$:

Thus $x=3/2$ or $x=18/5$. Each branch of $C$ is strictly increasing. The root $3/2$ lies to the left of the vertical asymptote $9/5$ and is the upper endpoint of the first interval; $18/5$ lies to its right and is the upper endpoint of the second interval.

Therefore $\boxed{p=3/2,\qquad q=18/5}$.