Math topic中文

Asymptotes symmetry restrictions

Equations of asymptotes, axes of symmetry, and restrictions on the possible values of $x$ and/or $y$ .

Practice questions

1The curve $C$ has equation $y = \frac{x^2 + x + 3}{x - 2}$ . Without using a calculator, find the set of values that $y$ …
25 (a) Solve the following integral. $\displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}}\,\mathrm{d}x.$ $\displaystyle\int x\ln\left(2-x^2\right)\,\mathrm{d}x$ …
3A curve $C_1$ has the equation $p^2x^2 - y^2 = p^2$ where $p > 1$ . Sketch $C_1$ , stating the coordinates of any points o…
4A curve with equation $y=f(x)$ is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
5The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
6Sketch, on the same axes, the graphs of $y=5-\dfrac{1}{x-1}$ and $y=\dfrac{1}{(x-5)^2}$ . Solve the inequality $5-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}$ …
7The curve $C$ has equation $y = \frac{x^2 + kx + 4}{x + 1}$ , where $k$ is a constant. Find the range of values of $k$ if…
8A curve C has equation $y = k(x+1) + \dfrac{4}{x+1}$ , $x \in \mathbb{R}$ , $x \ne -1$ , where $0 < k < 2$ is a constant. T…
9Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
10The diagram shows the graph of $y = f(x)$ . The curve has a turning point at $(0,a)$ and the lines $y = \pm kx$ are asymp…
11The curve $C$ has equation $x^2 - 4y^2 = 4$ . Sketch $C$ , stating the equations of any asymptotes and the coordinates of…
12A curve $y=f(x)$ undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation…
13A curve $C$ has equation $y = \frac{x^2 - 4x + 4}{x - 1}$ . Sketch $C$ , stating clearly the equations of any asymptotes,…
14The curve C is given by the equation $\dfrac{y^2}{x^2}=\dfrac{3a-x}{a+x},\quad x \ne 0,$ where $a$ is a positive const…
15Sketch the graph of $y = \frac{7-2x}{3x^2 + x - 14}$ . Give the equations of the asymptotes and the coordinates of the tu…
16Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where $|z+3|=2\,\Re(z)$ …
17The curves $C_1$ and $C_2$ have equations $y = \frac{2x^2 + 9}{x^2 - 4}$ and $\frac{y^2}{25} - \frac{x^2}{9} = 1$ respec…
18(Q3) [Cambridge/H2 Maths 2007] Show that the equation $y = \dfrac{2x + 7}{x + 2}$ can be written as $y = A + \dfrac{B}{x + 2}$ …
19In this question, $a$ , $b$ , $c$ and $d$ are non-zero constants. The curve $C$ has equation $y = \frac{ax^2 + bx - 7}{x - c}$ …
20The equation of a curve is $y=\dfrac{x^2-3a^2}{x-a}$ , where $a$ is a positive constant. Find the equations of the asympt…
21The curve $C_1$ has equation $y = \frac{ax^2+bx+3}{x+2}$ , where $a$ and $b$ are constants. It is given that $C_1$ has an…
22The function f is defined by $f: x \mapsto \dfrac{1}{1 - x^2}$ , $-1 < x \le 0$ . (i) Define $f^{-1}$ . (ii) Sketch the gra…
23The diagram shows the graph of $y = f(x)$ with asymptotes $y = k$ , $x = 0$ and the graph cuts the x-axis at $(a,0)$ . On…
24A curve C has equation $y = f(x)$ , where $f(x) = \frac{2(x-1)^2}{x-3}$ . Sketch C, showing clearly the equations of the a…
25It is given that $\mathrm{f}(x) = x + \frac{m^2}{x-2}$ , where $0 < m < 1$ . Sketch the graph of $y = \mathrm{f}(x)$ , show…
26A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1} \quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
27A curve $C$ is defined parametrically by $x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …
28A curve $C$ has equation $y = \frac{1}{\sqrt{bx^2 - 2bx}}$ , where $b < 0$ . Sketch $C$ and give, in terms of $b$ where ap…
29A freshly brewed cup of tea, initially at a temperature of $85^\circ\text{C}$ , is left out to cool in a room with a room…
30A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
31Find, in terms of $a$ , the roots of the equation $\dfrac{1}{(x-a)^2}=|x-a|$ . On the same axes, sketch the curves with eq…
32The variables $x$ and $y$ are related by the differential equation $\pi \dfrac{dy}{dx} + y(3 - \pi \tan x) = 0$ . Using t…
33The figure below shows the sketch of the graph $y = f(x)$ . The graph has asymptotes at $x = 1$ , $y = 3$ and the x-axis.…
34The diagram above shows a sketch of the graph of $y=f(x)$ for $x \\ge 0$ . There are $n$ rectangles each of width $h$ dra…
35Sketch the curve with equation $y = \ln(x+1), x > -1$ , stating the equation of asymptote and intercepts with the axes. O…
36The function $\text{f}$ is defined by $\text{f} : x \to 2x - \frac{1}{2x}, \quad 0 < x < 2.$ It is given that $\text{f}^{-1}$ …
37[In this question, sketches of the given graphs are not drawn to scale] The graphs of $y = |\mathrm{f}(x)|$ and $y^2 = \mathrm{f}(x)$ …
38The curve $C$ has equation $\frac{(x-1)^2}{9} - y^2 = 1.$ Sketch the curve $C$ , showing clearly the equations of asymp…
39The curve C has equation $y=\dfrac{2x^{2}-a}{x+k}$ where $a,\; k>0$ . Given that $2k^{2}\ne a$ , find the range of values…
40The curves $C_{1}$ and $C_{2}$ have equations $\dfrac{x^{2}}{9} - \dfrac{y^{2}}{4} = 1$ and $y^{2} + x^{2} = k^{2}$ resp…
41Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f} : x \to \frac{4}{(x-4)^2}, \quad x \in \mathbb{R}, x \ne 4,$ …
42(Q2) [Cambridge/9231/2007] The curve C has equation $y = \frac{ax^2 + bx + c}{x + 4},$ where $a$ , $b$ and $c$ are cons…
43The function f is defined by $f : x \mapsto \dfrac{5}{(x-4)^2}$ for $x\in\mathbb{R},\ x<4$ . Find $f^{-1}(x)$ and state t…
44The function f is defined by $f : x \mapsto 4 + \dfrac{1}{(x-1)^2 + 1},\quad x \ge 0$ . The function g is defined by g :…
45A curve has equation $y=f(x)$ , where $f(x)=x^{3}+ax^{2}+bx+c$ . The curve $y=f(x)$ has a maximum point at $(1,-32)$ , whil…
46It is given that the curve $C$ has equation $y = \dfrac{-4x^2 - 6x - 11}{2x + 2}$ , $x \in \mathbb{R}$ , $x \ne -1$ . Witho…
47The diagram below shows the graph of $y = \text{f}(x)$ . The curve passes through the $x$ -axis at $(-2, 0)$ and $(1, 0)$ ,…
48On the same axes, sketch the graphs of $y = \frac{1}{x - 2}$ and $y = 4x - 2$ , indicating clearly the equations of the a…
49The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{ax}{x-a}, \quad \text{for } x \in \mathbb{R}, \ -a \le x < a,$ …
50The curve $C$ has equation $y = \frac{x^2 + 4x + \lambda - 5}{x + r},$ where $\lambda$ is a non-zero constant, and a v…
51Sketch, on the same diagram, the curves with equations $y = \frac{|x + 1|}{|x - 2|}$ and $y = \ln\left(1 - \frac{x}{2}\right) + \tfrac{1}{2}$ …
52Sketch the curve $C$ with equation $y = \frac{x-2}{x-1}$ , stating the exact coordinates of any points of intersection wi…
53A curve has parametric equations $x=2t$ , $y=e^{-t^2}$ . Sketch the curve, stating the exact coordinates of any points of…
54The curve $C_1$ has equation $y = \dfrac{ax^2+bx+3}{x+2}$ , where $a$ and $b$ are constants. It is given that $C_1$ has a…
55The diagram shows the graph of $y = \text{f}'(x)$ , which passes through the origin and has turning points at $(2, 5)$ an…
56Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
57State a sequence of transformations that will transform the curve with equation $y^{2}-x^{2}=1$ on to the curve with equ…
58The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{2x}{x^2 - 1}, \quad x \in \mathbb{R}, \ x > 1$ . Sk…
59A function $\text{h}$ is self-inverse if $\text{h}(x) = \text{h}^{-1}(x)$ for all $x$ in the domain of $\text{h}$ . The f…
60The variables $x$ and $y$ are related by the differential equation $\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 2\mathrm{e}^{-2x} .$ …