FPM Asymptote graphs (Chapter 4)

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1. (2012/june/paper02/q7)

The curve $G$ has equation $y=3-\frac{1}{x-1}, x \neq 1$

(a) Find an equation of the asymptote to $G$ which is parallel to

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(b) Find the coordinates of the point where $G$ crosses

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(c) Sketch $G$, showing clearly the asymptotes and the coordinates of the points where the curve crosses the coordinate axes. ( 3 marks)

A straight line $l$ intersects $G$ at the points $P$ and $Q$. The $x$-coordinate of $P$ and the $x$-coordinate of $Q$ are roots of the equation $2 x-3=\frac{1}{x-1}$

(d) Find an equation of $l$. ( 2 marks)

2. (2013/june/paper01/q3)

Figure 1 shows a sketch of the curve with equation $y=1+\frac{c}{x+a}$, where $a$ and $c$ are integers.

The equations of the asymptotes to the curve are $x=3$ and $y=b$.

(a) Find the value of $a$ and the value of $b$. ( 2 marks)

The curve crosses the $x$-axis at $(1,0)$ and the $y$-axis at $(0, d)$.

(b) Find the value of $c$ and the value of $d$. ( 4 marks)

3. (2014/jan/paper02/q5)

A curve $C$ has equation $y=\frac{2 x-5}{x+3}, x \neq-3$

(a) Find an equation of the asymptote to $C$ which is parallel to

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(b) Find the coordinates of the point where $C$ crosses

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(c) Sketch the graph of $C$, showing clearly its asymptotes and the coordinates of the points where the graph crosses the coordinate axes. ( 3 marks)

(d) Find the gradient of $C$ at the point on $C$ where $x=-1$ ( 3 marks)

4. (2014/june/paper02/q8)

A curve has equation $y=\frac{3 x-2}{4 x+5}, \quad x \neq-\frac{5}{4}$

(a) Write down an equation of the asymptote to the curve which is parallel to

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(b) Find the coordinates of the point where the curve crosses

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(c) Sketch the curve, showing clearly the asymptotes and the coordinates of the points where the curve crosses the coordinate axes. ( 3 marks)

(d) Find an equation of the normal to the curve at the point where $x=-1$ Give your answer in the form $a x+b y+c=0$ where $a, b$ and $c$ are integers. ( 7 marks)

5. (2015/june/paper01/q9)

A curve $C$ has equation $y=\frac{3 x+1}{2 x+3} \quad x \neq-\frac{3}{2}$

(a) Write down an equation of the asymptote of $C$ which is parallel to

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(b) Find the coordinates of the points where $C$ crosses

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(c) Using the axes opposite, sketch the curve $C$, showing clearly the asymptotes and the coordinates of the points where $C$ crosses the axes.

The curve $C$ intersects the $x$-axis at the point $A$.

The line $/$ is the normal to $C$ at $A$. ( 3 marks)

(d) Find an equation for $l$. ( 5 marks)

The line $l$ meets $C$ again at the point $B$.

(e) Find the $x$-coordinate of $B$. ( 5 marks)

6. (2016/june/paper02/q8)

A curve $C$ has equation $$ y=\frac{3 x^2-1}{3 x+2} \quad \text { where } x \neq-\frac{2}{3} $$

(a) Write down an equation of the asymptote to $C$ which is parallel to the $y$-axis. ( 1 marks)

(b) Find the coordinates of the stationary points on $C$. ( 8 marks)

The curve crosses the $y$-axis at the point $A$.

(c) Write down the coordinates of $A$. ( 1 marks)

(d) On the axes on the opposite page, sketch $C$, showing clearly the asymptote parallel to the $y$-axis, the coordinates of the stationary points and the coordinates of $A$. ( 3 marks)

The line $l$ is the normal to the curve at $A$.

(e) Find an equation of $l$. ( 3 marks)

7. (2017/jan/paper01/q6)

Figure 3 shows a sketch of the curve with equation $$ y=\frac{b x+c}{x+a} \quad x \neq-a, $$

where $a, b$ and $c$ are integers.

The equations of the asymptotes to the curve are $x=-2$ and $y=3$

The curve crosses the $y$-axis at $(0,3.5)$

(a) Write down the value of $a$ and the value of $b$. ( 2 marks)

(b) Find the value of $c$. ( 2 marks)

Given that the curve crosses the $x$-axis at $(s, 0)$

(c) find the value of $s$. ( 2 marks)

8. (2017/june/paper01/q10)

A curve $C$ has equation $y=8 x+\frac{1}{2 x-1} \quad x \neq \frac{1}{2}$

(a) Write down an equation of the asymptote to $C$ which is parallel to the $y$-axis. ( 1 marks)

(b) Show that $C$ has a minimum point at $x=\frac{3}{4}$ and a maximum point at $x=\frac{1}{4}$ ( 9 marks)

(c) Find the $y$ coordinate of

(i) the minimum point,

(ii) the maximum point,

(iii) the point where $C$ crosses the $y$-axis. ( 3 marks)

(d) Sketch the curve $C$, showing clearly the asymptote found in part (a), the coordinates of the turning points and the coordinates of the point where $C$ crosses the $y$-axis. ( 3 marks)

9. (2018/june/paper01/q6)

The curve $C$ has equation $y=\frac{2 x-4}{x-3} \quad x \neq 3$

(a) Write down an equation of the asymptote to $C$ which is parallel to

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(b) Find the coordinates of the point where $C$ crosses

(i) the $x$-axis,

(ii) the $y$-axis. ( 2 marks)

(c) Sketch $C$, showing clearly the asymptotes and the coordinates of the points where $C$ crosses the coordinate axes. ( 3 marks)

10. (2019/juneR/paper02/q2)

Figure 1 shows part of the curve $S$ with equation $y=\frac{a x+b}{x+c}$ where $a, b$ and $c$ are integers.

The asymptote to $S$ that is parallel to the $x$-axis has equation $y=-2$

The asymptote to $S$ that is parallel to the $y$-axis has equation $x=-3$

The curve crosses the $x$-axis at the point with coordinates $(4,0)$

The curve crosses the $y$-axis at the point with coordinates $(0, p)$ where $p$ is a rational number.

Find

(i) the value of $a$,

(ii) the value of $b$,

(iii) the value of $c$,

(iv) the value of $p$. ( 4 marks)

Answers

1.(a)(i) $y=3$ (ii) $x=1 \quad$ (b) $\left(\frac{4}{3}, 0\right),(0,4)$ (c) Graph (d) $y=6-2 x$

2.(a) $a=-3, b=1$ (b) $c=2, d=\frac{1}{3}$

3. $(a)$ (i) $y=2$ (ii) $x=-3$ (b) $\left(\frac{5}{2}, 0\right),\left(0,-\frac{5}{3}\right)$ (c) Graph (d) $\frac{11}{4}$

4.(a) (i) $y=\frac{3}{4}$ (ii) $x=-\frac{5}{4}$ (b) (i) $\left(\frac{2}{3}, 0\right)\left(\right.$ ii) $\left(0,-\frac{2}{5}\right)$ (c) Graph (d) $x+23 y+116=0$

5.(a) (i) $y=\frac{3}{2}$ (ii) $x=-\frac{3}{2}$ (b) (i) $x=-\frac{1}{3}$ (ii) $y=\frac{1}{3}$ (c) Graph (d) $y=-\frac{7}{9}\left(x+\frac{1}{3}\right)$ (e) $-\frac{24}{7}$

6.(a) $x=-\frac{2}{3}$ (b) $(-1,-2),\left(-\frac{1}{3},-\frac{2}{3}\right)$ (c) $A=\left(0,-\frac{1}{2}\right)$ (d) Graph (e) $y=-\frac{4}{3} x-\frac{1}{2}$

7.(a) $a=2,b=3$ (b) $c=7$ (c) $s=-\frac{7}{3}$

8.(a) $x=\frac{1}{2}$ (b) Show(c)(i) $y=8 \quad$ (ii) $y=0$ (iii) $y=-1$ (d) Figure

9.(a) (i) $y=2$ (ii) $x=3$ (b)(i) (2,0) (ii) (0,$\frac{4}{3}$) (c) Graph

10.(i) $a=-2$ (ii) $b=8$ (iii) $c=3$ (iv) $p=\frac 83$