Math topic中文

Characteristics of standard graphs

Important characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes of the following:

$y^2 = ax$
$x^2 = by$
$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ and $\dfrac{y^2}{b^2} - \dfrac{x^2}{a^2} = 1$
$y = \dfrac{ax + b}{cx + d}$
$y = \dfrac{ax^2 + bx + c}{dx + e}$

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 $y$ …
2The functions $f$ and $g$ are defined by $f : x \mapsto \sqrt{16 - 4x},\; x \in \mathbb{R},\; x \le 4,$ $g : x \mapsto x^{2},\; x \in \mathbb{R}.$ …
3By expressing the equation of the curve $y = \frac{12x + 11}{2x + 1}$ in the form $y = A + \frac{B}{2x + 1}$ , where $A$ …
45 (a) Solve the following integral. $\displaystyle\int \frac{1-2x}{\sqrt{3+2x-x^2}}\,\mathrm{d}x.$ $\displaystyle\int x\ln\left(2-x^2\right)\,\mathrm{d}x$ …
5A 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…
6Sketch, on the same diagram, the graphs of $y = |2x + 3|$ and $y = -2x^2 - 8x - 3$ . (You are not required to label any a…
7The curve $C_1$ with equation $\frac{x^2}{a^2}+\frac{y^2}{4a^2}=1$ is transformed to curve $C_2$ by a translation of $a$ …
8The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
9Sketch, on the same axes, the graphs of $y=5-\dfrac{1}{x-1}$ and $y=\dfrac{1}{(x-5)^2}$ . Solve the inequality $5-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}$ …
10The 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…
11A 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…
12Find the exact roots of the equation $\left|\dfrac{4x-5}{x-2}\right| = 3x. On the same axes, sketch the curves with equations$ …
13The curve $C$ has equation $x^2 - 4y^2 = 4$ . Sketch $C$ , stating the equations of any asymptotes and the coordinates of…
14A curve $y=f(x)$ undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation…
15The diagram shows an ellipse, centred at the origin with semi-minor axis $a$ and semi-major axis $2a$ , where $a > 0$ . Th…
16A curve $C$ has equation $y = \frac{x^2 - 4x + 4}{x - 1}$ . Sketch $C$ , stating clearly the equations of any asymptotes,…
17The function f, with domain the set of all real values, is given by $\text{f}(x) = \begin{cases} 4 - x^2 & \text{for } 0 < x \le 2, \\ 2x - 4 & \text{for } 2 < x \le 4, \end{cases}$ …
18Section A: Pure Mathematics [60 marks] The curve $C$ has equation $y = \frac{x^2 - 2x + 4}{x - b}$ , $x \ne b$ . Given tha…
19The diagram below shows a parabola which cuts the $x$ -axis at $(-2a, 0)$ , $(2a, 0)$ and has a maximum point at $(0, 4a^2)$ …
20Sketch the graph of $y = \frac{7-2x}{3x^2 + x - 14}$ . Give the equations of the asymptotes and the coordinates of the tu…
21Find, in a simplified form, the cartesian equation of the locus of the point P representing the complex number z where $|z+3|=2\,\Re(z)$ …
22The 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…
23(Q3) [Cambridge/H2 Maths 2007] Show that the equation $y = \dfrac{2x + 7}{x + 2}$ can be written as $y = A + \dfrac{B}{x + 2}$ …
24A function f is defined as $f(x)=\ln(5-x),\; x<5.$ Find the volume of revolution when the region bounded by the curve of…
25The figure below shows a sketch of part of the graph of $y = f(x)$ for $0 \\le x < 2a$ . The vertical asymptote $x = 2a$ …
26(Q1) [Cambridge/9794/2007] The equation of a circle is $x^2 + y^2 - 8y = 9$ . Find the coordinates of the centre of the c…
27In this question, $a$ , $b$ , $c$ and $d$ are non-zero constants. The curve $C$ has equation $y = \frac{ax^2 + bx - 7}{x - c}$ …
28The 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…
29Sketch the curve with equation $\frac{x^2}{400} + \frac{(y-10)^2}{900} = 1.$ [There is no need to indicate the x-inter…
30A 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…
31It 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…
32A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1} \quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
33A curve $C$ has equation $y = \frac{1}{\sqrt{bx^2 - 2bx}}$ , where $b < 0$ . Sketch $C$ and give, in terms of $b$ where ap…
34A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
35The function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \begin{cases} 4 - x & \text{for } 1 \le x < 3, \\ (x - 4)^2 & \text{for } 3 \le x < 4, \end{cases}$ …
36It is given that $f(x)=\begin{cases}4x,&0\le x<1\\(3-x)^2,&1\le x\le 3\end{cases}$ and $f(-x)=-f(x)$ . It is also kno…
37Find, 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…
38Sketch the curve $C$ with equation $x^2 - (y + 1)^2 = 1$ , indicating clearly the coordinates of any points of intersecti…
39In harvesting of renewable natural resources, it is desirable that policies are formulated to allow maximal harvest of t…
40Sketch the curve with equation $y = \ln(x+1), x > -1$ , stating the equation of asymptote and intercepts with the axes. O…
41The curve $C$ has equation $\frac{(x-1)^2}{9} - y^2 = 1.$ Sketch the curve $C$ , showing clearly the equations of asymp…
42The 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…
43The 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…
44Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f} : x \to \frac{4}{(x-4)^2}, \quad x \in \mathbb{R}, x \ne 4,$ …
45(Q2) [Cambridge/9231/2007] The curve C has equation $y = \frac{ax^2 + bx + c}{x + 4},$ where $a$ , $b$ and $c$ are cons…
46The 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…
47The curve $C$ is defined by the parametric equations $x = t^2$ and $y = 2t + t^3$ , where $-1 \le t \le 1$ . Sketch the gr…
48The 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 :…
49It is given that the curve $C$ has equation $y = \dfrac{-4x^2 - 6x - 11}{2x + 2}$ , $x \in \mathbb{R}$ , $x \ne -1$ . Witho…
50On the same axes, sketch the graphs of $y = \frac{1}{x - 2}$ and $y = 4x - 2$ , indicating clearly the equations of the a…
51The function $\mathrm{f}$ is defined by $\mathrm{f} : x \mapsto \frac{ax}{x-a}, \quad \text{for } x \in \mathbb{R}, \ -a \le x < a,$ …
52The curve $C$ has equation $y = x\cos 2x$ , where $0 \le x \le \pi$ . Find the exact $x$ -coordinates of the points where $C$ …
53The curve $C$ has equation $y = \frac{x^2 + 4x + \lambda - 5}{x + r},$ where $\lambda$ is a non-zero constant, and a v…
54Sketch, on the same diagram, the curves with equations $y = \frac{|x + 1|}{|x - 2|}$ and $y = \ln\left(1 - \frac{x}{2}\right) + \tfrac{1}{2}$ …
55Sketch the curve $C$ with equation $y = \frac{x-2}{x-1}$ , stating the exact coordinates of any points of intersection wi…
56A curve has parametric equations $x=2t$ , $y=e^{-t^2}$ . Sketch the curve, stating the exact coordinates of any points of…
57Section A: Pure Mathematics [40 marks] Sketch, on the same axes, the graphs of $y = 3x$ and $y = |x^2 - a^2|$ where $a > 0$ …
58Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
59State a sequence of transformations that will transform the curve with equation $y^{2}-x^{2}=1$ on to the curve with equ…
60A 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…