曲线 $C$ 的方程为 $y=\dfrac{2x^2-5x}{2x^2-7x-4}$。

(a) 求 $C$ 的渐近线方程。

(b) 求 $C$ 上所有驻点的坐标。

(c) 画出 $C$ 的草图，并写出它与坐标轴交点的坐标。

(d) 画出方程为 $y=\left|\dfrac{2x^2-5x}{2x^2-7x-4}\right|$ 的曲线的草图。

(e) 求满足 $\left|\dfrac{2x^2-5x}{2x^2-7x-4}\right|<\dfrac19$ 的 $x$ 的取值集合，用精确形式表示。

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。

分母因式分解给出两条竖直渐近线，最高次项系数比给出水平渐近线。

The denominator factorises as

Neither factor cancels the numerator, so the vertical asymptotes are $\boxed{x=-1/2}$ and $\boxed{x=4}$.

The ratio of leading coefficients is $1$, giving the horizontal asymptote $\boxed{y=1}$.

用商法则求导并因式分解分子，分别代回函数求两个驻点。

Put $N=2x^2-5x$ and $D=2x^2-7x-4$. By the quotient rule,

Thus $x=-5$ or $x=1$. Substitution gives

The derivative changes from negative to positive at $-5$ and from positive to negative at $1$, so these are a local minimum and local maximum, respectively.

除两条竖直渐近线外，还要标明曲线与水平渐近线的交点，左支的极小值略低于1。

The axis intersections are $\boxed{(0,0)}$ and $\boxed{(5/2,0)}$. Also

so the curve crosses $y=1$ at $(-2,1)$.

Sketch three branches, labelling $x=-1/2$, $x=4$ and $y=1$:

• On $x<-1/2$, the branch approaches $1$ from below as $x\to-\infty$, has a minimum at $(-5,25/27)$, crosses $(-2,1)$ and tends to $+\infty$ as $x\to(-1/2)^-$.
• On $-1/2<x<4$, it tends to $-\infty$ at both ends, crosses $(0,0)$, rises to the maximum $(1,1/3)$, then falls through $(5/2,0)$.
• On $x>4$, it decreases from $+\infty$ towards $1$ from above.

把两个横轴下方的区间反射到上方，两个原横轴交点都形成尖点。

Reflect the portions with $-1/2<x<0$ and $5/2<x<4$ in the $x$-axis. Leave the remaining portions unchanged.

The transformed graph lies on or above the $x$-axis, has sharp minima at $(0,0)$ and $(5/2,0)$, and retains the vertical asymptotes $x=-1/2$, $x=4$ and horizontal asymptote $y=1$.

先求函数值为正负1/9的四个临界点，再通过平方不等式的乘积符号选出两段区间。最后检查不包含分母零点。

Write $N=2x^2-5x$ and $D=2x^2-7x-4\ne0$. The boundary equations are

Their roots are, respectively,

Since both sides of $|N/D|<1/9$ are non-negative, squaring and multiplying by $81D^2>0$ is valid. It gives

or

The four simple roots are ordered $b_-<a_-<a_+<b_+$. The product has positive leading coefficient, so it is negative between the first two and the last two roots. Neither interval contains $x=-1/2$ or $x=4$.

Therefore

or