Math topic中文

Transformations of f

Effect of transformations on the graph of $y = \mathrm{f}(x)$ as represented by $y = a\,\mathrm{f}(x)$ , $y = \mathrm{f}(x) + a$ , $y = \mathrm{f}(x + a)$ and $y = \mathrm{f}(ax)$ , and combinations of these transformations.

Practice questions

1将函数 $f(x)=\sin 2x$ 的图象保持纵坐标不变，先将横坐标缩短为原来的 $\dfrac{1}{2}$ ，再向右平移 $\dfrac{π}{6}$ 个单位长度后得到 $g(x)$ ，则 $g(x)$ 的解析式为 $(\quad)$ …
2By expressing the equation of the curve $y = \frac{12x + 11}{2x + 1}$ in the form $y = A + \frac{B}{2x + 1}$ , where $A$ …
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…
4The diagram shows the graphs of $y = \mathrm{f}(|x|)$ and $y = \mathrm{f}'(x)$ . The graph of $y = \mathrm{f}(|x|)$ has t…
5The 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 $a$ …
6A curve with equation $y=f(x)$ is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
7The diagram shows the curve $y = \text{f}(1-x)$ . The curve has a maximum point at $(-1, 5)$ and two asymptotes $x = -2$ …
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…
9The 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…
10The curve $C$ has equation $x^2 - 4y^2 = 4$ . Sketch $C$ , stating the equations of any asymptotes and the coordinates of…
11A curve $y=f(x)$ undergoes in succession, the following transformations: A: A reflection in the y-axis B: A translation…
12The function f, with domain the set of all real values, is given by $\text{f}(x) = \begin{cases} 4 - x^2 & \text{for } 0 < x \le 2, \\ 2x - 4 & \text{for } 2 < x \le 4, \end{cases}$ …
13Section A: Pure Mathematics [60 marks] The curve $C$ has equation $y = \frac{x^2 - 2x + 4}{x - b}$ , $x \ne b$ . Given tha…
14(Q3) [Cambridge/H2 Maths 2007] Show that the equation $y = \dfrac{2x + 7}{x + 2}$ can be written as $y = A + \dfrac{B}{x + 2}$ …
15The figure below shows a sketch of part of the graph of $y = f(x)$ for $0 \\le x < 2a$ . The vertical asymptote $x = 2a$ …
16The 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…
17The 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…
18It 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…
19It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
20A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1} \quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
21要得到函数 $y=\sin 2x$ 的图象，只需将函数 $y=\sin (2x- \dfrac {π}{3})$ 的图象 $()$ A. 向右平移 $\dfrac {π}{6}$ 个单位长度 B. 向左平移 $\dfrac {π}{6}$ …
22A function $\text{f}$ is defined by $\text{f}(x) = x^3 + ax^2 + bx + c$ , where $a$ , $b$ and $c$ are constants. The graph…
23The function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \begin{cases} 4 - x & \text{for } 1 \le x < 3, \\ (x - 4)^2 & \text{for } 3 \le x < 4, \end{cases}$ …
24It 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…
25Sketch the curve $C$ with equation $x^2 - (y + 1)^2 = 1$ , indicating clearly the coordinates of any points of intersecti…
26The figure below shows the sketch of the graph $y = f(x)$ . The graph has asymptotes at $x = 1$ , $y = 3$ and the x-axis.…
27Describe a sequence of two transformations which maps the graph of $y = e^{x}$ onto the graph of $y = -e^{x} + 2$ . The l…
28[In this question, sketches of the given graphs are not drawn to scale] The graphs of $y = |\mathrm{f}(x)|$ and $y^2 = \mathrm{f}(x)$ …
29The 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…
30(Q2) [Cambridge/9231/2007] The curve C has equation $y = \frac{ax^2 + bx + c}{x + 4},$ where $a$ , $b$ and $c$ are cons…
31It is given that the curve $C$ has equation $y = \dfrac{-4x^2 - 6x - 11}{2x + 2}$ , $x \in \mathbb{R}$ , $x \ne -1$ . Witho…
32The diagram below shows the graph of $y = \text{f}(x)$ . The curve passes through the $x$ -axis at $(-2, 0)$ and $(1, 0)$ ,…
33将抛物线 $y=x^{2}$ 向左平移 $3$ 个单位，再向下平移 $4$ 个单位，平移后所得新抛物线的表达式为 $(\quad)$ A. $y=(x-3)^{2}+4$ B. $y=(x+3)^{2}+4$ C. $y=(x-3)^{2}-4$ …
34Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
35State a sequence of transformations that will transform the curve with equation $y^{2}-x^{2}=1$ on to the curve with equ…
36A heated metal block at $160^\circ\text{C}$ is left to cool in a laboratory with an ambient temperature of $30^\circ\text{C}$ …
37Sketch the graph of $(y+2)^2 = 4(x-1)^2 - 9$ , showing its main relevant features. Hence find the range of values of $k$ …
38A curve $C$ undergoes, in succession, the following transformations. $A$ : Translation of 4 units in the negative $y$ -dir…
39Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \quad \text{and} \quad y = b|x - a|.$ …
40Find the area of the region bounded by the graphs of $y = 2x^2 + 3$ and $y = -2x + 7$ . State the area of the region boun…
41The graph of a function f(x) has undergone the following transformations: Step 1: Translation in the negative x-directio…
42On the same axes, sketch the graphs of $y = \sin 2x$ and $y = \cos x$ for $0 \le x \le \frac{\pi}{2}$ , stating the exact…
43The curve $C_1$ with equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{4a^2}=1$ is transformed to curve $C_2$ by a translation of $a$ …
44若将函数y=3x+6沿x轴向右平移3个单位，则所得函数解析式为．
45A curve $R$ has equation $y=f(x)$ . The point $P$ with coordinates $(7,-9)$ lies on $R$ . The tangent to $R$ at point $P$ …
46如果将一次函数y=3x-1的图象沿y轴向上平移5个单位，那么平移后所得图象的函数解析式为___．
47The diagram shows the graphs of $y = f(|x|)$ and $y = f'(x)$ . The graph of $y = f(|x|)$ has turning points at $(-3,4)$ a…
48The graph of $y = f(x)$ is given below. On separate diagrams, sketch the graphs of $y = \frac{1}{f(x)}$ $y = f'(x).$ A c…
49The diagram shows the curve $y = \mathrm{f}(x)$ . The curve passes through the point $A(a,0)$ and the point $B(b,0)$ , has…
50The curve $C$ defined by $y = \frac{ax + 1}{bx - 2}$ passes through $(3, 13)$ and has vertical asymptote $x = 2$ . Find t…
51The diagram below shows the graph of $y = \mathrm{f}(x)$ . The asymptotes of the graph are $x = 2$ and $y = 2$ . The graph…
52A heated metal block at $160^\circ$ C is left to cool in a laboratory with an ambient temperature of $30^\circ$ C. The te…
53设 $f(x)=2 \sqrt{3}\sin (π-x)\sin x-(\sin x-\cos x{)}^{2}$ ． $(I)$ 求 $f(x)$ 得单调递增区间； $(II)$ 把 $y=f(x)$ 的图象上所有点的横坐标伸长到…
54The above diagram shows the curve $C$ with equation $e^{2y} = x^2 + y$ for $x>0$ . Make a copy of the diagram to include…
55The diagram below shows the graph of curve $C_1$ , with equation $y = \mathrm{f}(x)$ , defined for $x \in \mathbb{R} \setminus \{4\}$ …
56A curve $C_1$ has parametric equation $x = 3\sec^2 \theta$ , $y = 2\tan \theta$ , $0 \le \theta < \frac{\pi}{2}$ . $L_1$ an…
57The diagram shows the curve $y = \text{f}(x)$ . The curve passes through the points $A(-1, 0)$ , $B(0, 2)$ , $C(1, 0)$ and…
58The diagram shows the curve $y = f(x)$ , with turning points at $A(3, 2.5)$ and $B(0, 1)$ . The lines $y = 1$ and $x = -2$ …
59A curve $C_1$ is defined parametrically by $x = \frac{2}{t-1}\quad\text{and}\quad y = \frac{4}{t+1},\qquad t \ne \pm 1.$ …
60The diagram below shows the graph of $y = \mathrm{f}(x)$ . The curve cuts the axes at $A(0, 1.5)$ and $B(-3, 0)$ . The asy…