Math topic中文

Graphs

graphs of y=asin⁡(bx)+cy = a\sin(bx) + c, y=asin⁡ ⁣(xb)+cy = a\sin\!\left(\frac{x}{b}\right) + c, y=acos⁡(bx)+cy = a\cos(bx) + c, y=acos⁡ ⁣(xb)+cy = a\cos\!\left(\frac{x}{b}\right) + c and y=atan⁡(bx)y = a\tan(bx), where aa is real, bb is a positive integer and cc is an integer

Practice questions

  1. 1State the range of principal values of tan⁡−1x\tan^{-1} x. The curve y=a+bcos⁡cxy = a + b \cos c x is shown below for 0≤x≤3π20 \le x \le \tfrac{3\pi}{2}…
  2. 2The diagram below shows a portion a plastic fibre cable, which allows light waves to pass through. The path of the light…
  3. 3The diagram shows a wind turbine with blade 25 m25\text{ m} in length. The centre of the wind turbine is 35 m35\text{ m} fro…
  4. 4The diagram shows the curve y=acos⁡bx+cy = a\cos bx + c for 0≤x≤3π20 \le x \le \tfrac{3\pi}{2} radians. A minimum point (π2,−2)(\tfrac{\pi}{2}, -2)…
  5. 5A clock is set on the vertical wall on the roof of a building. The distance from the centre of the clock, OO, to the ti…
  6. 6The function f is defined by f(x)=1+2cos⁡2xf(x)=1+2\cos 2x for 0≤x≤π0\le x\le\pi. State the amplitude and the period of f(x)f(x). State t…
  7. 7The diagram shows the curve y=psin⁡xq+ry = p \sin \frac{x}{q} + r for 0≤x≤8π0 \le x \le 8\pi radians. The curve has a minimum point a…
  8. 8The function ff is defined, for x≥0∘x \ge 0^\circ, by f(x)=3cos⁡(2x3)−1f(x) = 3\cos\left(\frac{2x}{3}\right) - 1. State the maximum and…
  9. 9The height of water in a harbour at time tt hours after midnight on a certain day is modelled by the formula h=4.8sin⁡kπt+5.1h = 4.8 \sin k\pi t + 5.1…
  10. 10The diagram shows part of the graph of y=atan⁡(bx+c)y = a\tan(bx + c). The graph has vertical asymptotes at x=−4πx = -4\pi and x=4πx = 4\pi…
  11. 11The equation of a curve is y=3−4sin⁡2xy = 3 - 4\sin 2x. State the minimum and maximum value of yy. Sketch the curve y=3−4sin⁡2xy = 3 - 4\sin 2x…
  12. 12The equation of a curve is y=4−3cos⁡2xy = 4 - 3\cos 2x. State the period and amplitude of yy. Sketch the graph of y=4−3cos⁡2xy = 4 - 3\cos 2x…
  13. 13The function ff is defined by f(x)=4cos⁡ax+bf(x) = 4 \cos ax + b for 0≤x≤π0 \le x \le \pi, where aa and bb are constants. The period…
  14. 14The graph of y=5sin⁡3x+cy = 5 \sin 3x + c, where cc is a constant, passes through the point (50∘,12)(50^\circ, \frac{1}{2}). Show that…
  15. 15State the amplitude and period of 2sin⁡3x+12\sin 3x + 1. Sketch the graph of y=2sin⁡3x+1y = 2\sin 3x + 1 for 0≤x≤π0 \le x \le \pi. By drawi…
  16. 16The diagram shows the curve y=acos⁡bx+cy = a\cos bx + c for 0≤x≤2π30 \le x \le \dfrac{2\pi}{3} radians, where aa, bb and cc are int…
  17. 17The diagram shows the curve y=a+6cos⁡bxy = a + 6\cos bx for 0≤x≤2π30 \le x \le \frac{2\pi}{3}, where aa and bb are positive constant…
  18. 18It is given that f(x)=2−3sin⁡3x\text{f}(x) = 2 - 3\sin 3x and g(x)=4cos⁡(x2)−1\text{g}(x) = 4\cos\left(\frac{x}{2}\right) - 1. State the least and…
  19. 19The function f is defined by f(x)=4cos⁡ax+bf(x)=4\cos ax + b for 0≤x≤3π0 \le x \le 3\pi, where aa and bb are constants. The maximum va…
  20. 20Solve the equation 4cos⁡2A=3−2sin⁡A4\cos 2A = 3 - 2\sin A for 0≤A≤2π0 \le A \le 2\pi. It is given that f(x)=2cos⁡6x−12\text{f}(x) = 2\cos 6x - \frac{1}{2}…
  21. 21The diagram shows part of the graph of y=psin⁡(qx)+ry = p\sin(qx) + r. Find the value of each of the constants pp, qq and rr. Gi…
  22. 22Sketch, on the axes below, the graphs of y=−3sin⁡xy = -3\sin x and y=1−2xπy = 1 - \frac{2x}{\pi} for 0≤x≤2π0 \le x \le 2\pi. Find the nu…
  23. 23State the amplitude and period of the graph of y=3sin⁡x2+1y=3\sin\frac{x}{2}+1. Sketch the graph of y=3sin⁡x2+1y=3\sin\frac{x}{2}+1 for 0≤x≤4π0 \le x \le 4\pi…
  24. 24Sketch the graph of y=−sin⁡x+1y = -\sin x + 1 and y=3cos⁡2xy = 3\cos 2x for 0∘≤x≤360∘0^\circ \le x \le 360^\circ on the same axes. Hence, stat…
  25. 25It is given that y1=sin⁡x−2y_1 = \sin x - 2 and y2=−3cos⁡2xy_2 = -3 \cos 2x. State the amplitude and period, in degrees, of (a) y1y_1, (b…
  26. 26已知函数f(x)=(asin⁡x+cos⁡x)cos⁡x−12f(x)=(a\sin x+\cos x)\cos x-\dfrac{1}{2}的图象的一条对称轴为x=π6x=\dfrac{π}{6}，则下列结论中正确的是()(\quad) A. (−7π12,0)(-\dfrac{7π}{12},0)…
  27. 27The diagram shows part of the graph of y=asin⁡xb+cy = a \sin \frac{x}{b} + c. The graph has a maximum point at (6π,5)(6\pi, 5) and a…
  28. 28It is given that f(x)=2sin⁡x2f(x)=2\sin\frac{x}{2} and g(x)=3cos⁡x+1g(x)=3\cos x+1, where 0≤x≤2π0 \le x \le 2\pi. State the period of f(x)f(x). Sta…
  29. 29已知函数f(x)=(log⁡2x)2−log⁡2x2+3f\left(x\right)={\left({\log }_{2}x\right)}^{2}-{\log }_{2}{x}^{2}+3 ，当x∈[1,4]x∈\left[1,4\right] 时，f(x)f\left(x\right) …
  30. 30It is given that f(x)=2x3+ax2+x+b\mathrm{f}(x) = 2x^3 + ax^2 + x + b. Find the value of aa and of bb for which 2x2+x−12x^2 + x - 1 is a f…
  31. 31Sketch the graphs of y=−2cos⁡3x+1y=-2\cos 3x+1 and y=−2πx+1y= -\dfrac{2}{\pi}x+1 on the same axes, for 0≤x≤π0 \le x \le \pi. Explain how th…
  32. 32In Singapore, Alternating Current (AC) power is provided by utility companies to households at a frequency of 50 Hz, or…
  33. 33Given that y=psin⁡2x+qy = p \sin 2x + q, where pp and qq are positive integers. The maximum and minimum values of yy are 8 and…
  34. 34The function ff is defined, for 0≤x≤2π0 \le x \le 2\pi, by f(x)=1−2cos⁡2xf(x) = 1 - 2 \cos 2x. State the period and amplitude of ff. S…
  35. 35Find the amplitude and period of 3sin⁡sin⁡2x−23\sin\sin 2x - 2. Sketch the curve y=3sin⁡sin⁡2x−2y = 3\sin\sin 2x - 2 for 0∘≤x≤360∘0^\circ \le x \le 360^\circ…