Math topic中文

Completing the square

finding the maximum or minimum value of a quadratic function using the method of completing the square

Practice questions

  1. 1若f(x)=3x2−x+1f(x)=3x^{2}-x+1，g(x)=2x2+x−1g(x)=2x^{2}+x-1，则f(x)f(x)，g(x)g(x)的大小关系是()(\quad) A. f(x)=g(x)             f(x)=g(x)\:\:\:\:\:\:\:\:\:\:\:\:\: …
  2. 2A curve has the equation y=2x−33x+4y = \frac{2x - 3}{3x + 4}, where x≠−43x \ne -\frac{4}{3}. The normal to the point PP on the cur…
  3. 3对于函数f(x)=x2+2xf(x)=x^{2}+2x，在使f(x)⩾Mf(x)\geqslant M成立的所有实数MM中，我们把MM的最大值Mmax=−1M_{max}=-1叫做函数f(x)=x2+2xf(x)=x^{2}+2x的下确界，则对于a∈Ra∈R，…
  4. 4Express y=−2x2−12x+1y = -2x^2 - 12x + 1 in the form y=a−2(x+b)2y = a - 2(x + b)^2 and hence state the maximum value of yy, and its corresp…
  5. 5已知函数f(x)=x2−ax(a≠0)f(x)=x^{2}-ax(a\neq 0)，g(x)=ln⁡xg(x)=\ln x，f(x)f(x)图象与xx轴异于原点的交点MM处的切线为l1l_{1}，g(x−1)g(x-1)与xx轴的交点NN处的切线为…
  6. 6The equation of a curve is y=16−ax−x2y=16-ax-x^{2}, where aa is positive constant. Given that yy can be expressed in the form…
  7. 7Solve the equation 2x+1+2−x=32^{x+1}+2^{-x}=3. The equation of a curve is y=2x2−4ax+by=2x^2-4ax+b where aa and bb are non-zero constant…
  8. 8The equation of a curve is y=−2x2+3x+5y = -2x^2 + 3x + 5. Write y=−2x2+3x+5y = -2x^2 + 3x + 5 in the form y=a(x−h)2+ky = a(x - h)^2 + k, where aa,…
  9. 9The diagram shows the line y=24y = 24 and part of the graph of y=∣x2−16x+39∣y = |x^2 - 16x + 39|. The graph crosses the line y=24y = 24…
  10. 10某厂家生产一批遮阳伞，每个遮阳伞的成本价是2020元，试销售时发现：遮阳伞每天的销售量y(y(个))与销售单价x(x(元))之间是一次函数关系，当销售单价为2828元时，每天的销售量为260260个；当销售单价…
  11. 11Complete the table of values for y=x2+2x−1y = x^2 + 2x - 1 On the grid opposite, draw the graph of y=x2+2x−1y = x^2 + 2x - 1 for −4≤x≤2-4 \le x \le 2…
  12. 12不论aa，bb为何有理数，a2+b2−2a−4b+ca^{2}+b^{2}-2a-4b+c的值总是非负数，则cc的最小值是()(\quad) A. 44 B. 55 C. 66 D. 无法确定
  13. 13用配方法将函数y=12x2−2x+1y= \dfrac {1}{2}x^{2}-2x+1写成y=a(x−h)2+ky=a(x-h)^{2}+k的形式是()() A. y=12(x−2)2−1y= \dfrac {1}{2}(x-2)^{2}-1 B. y=12(x−1)2−1y= \dfrac {1}{2}(x-1)^{2}-1…
  14. 14抛物线y=x2−6x+5y=x^{2}-6x+5的顶点坐标为___.
  15. 15函数y=(x−2)(3−x)y=(x-2)(3-x)取得最大值时，x=x= ___ ．
  16. 16A ball is thrown vertically upwards. Its height hh m, above the ground at time tt seconds after being thrown is given…
  17. 17Express 3−3x−2x23 - 3x - 2x^2 in the form a(x+b)2+ca(x + b)^2 + c and hence state the coordinates of the turning point of the curve y=3−3x−2x2y = 3 - 3x - 2x^2…
  18. 18已知函数f(x)=x2+ax+b−a(a,b∈R)f(x)=x^{2}+ax+b-a(a,b∈R)． (1)若关于xx的不等式f(x)>0f(x) > 0的解集为(−∞,−1)∪(3,+∞)(-∞,-1)∪(3,+∞)，求实数aa，bb的值； (2)设a=2a=2，若不等式f(x)>b2−3bf(x) > b^{2}-3b…
  19. 19关于xx的方程x2−mx+m−2=0x^{2}-mx+m-2=0，对其根的情况叙述正确的是()() A. 有两个相等的实数根 B. 有两个不相等的实数根 C. 没有实数根 D. 根的情况不能确定
  20. 20Express −9+6x−2x2-9 + 6x - 2x^2 in the form a(x+b)2+ca(x+b)^2 + c and hence explain why −9+6x−2x2-9 + 6x - 2x^2 is negative for all real valu…
  21. 21Express 4x2+8x−54x^{2}+8x-5 in the form p(x+q)2+rp(x+q)^{2}+r, where pp, qq and rr are constants to be found. Hence, state the coo…
  22. 22Express y=4x−4x2−3y = 4x - 4x^2 - 3 in the form p(x+q)2+rp(x + q)^2 + r where pp, qq and rr are constants. Hence, explain whether 4x−4x2−3=04x - 4x^2 - 3 = 0…
  23. 23The cost per submarine, $y in millions, of assembling xx submarines can be modelled by y=52x2−20x+100y = \frac{5}{2}x^2 - 20x + 100…
  24. 24The diagram shows the cross section of a tunnel for vehicles to pass. The shape of the arched tunnel can be modelled by…
  25. 25Express each of 4x2−6x−14x^2 - 6x - 1 and −x2−10x+3-x^2 - 10x + 3 in the form a(x+b)2+ca(x+b)^2 + c, where aa, bb and cc are constants. U…
  26. 26已知二次函数y=x2−kx−8y={{x}^{2}}-kx-8过点P(2,−8)P(2,-8)，则当−1<x<2-1 < x < 2时，yy的取值范围是 ．
  27. 27Show that x2−10x+21x^2 - 10x + 21 can never be smaller than −4-4. Given that y=x2−10x+21y = x^2 - 10x + 21, write down the coordinates o…
  28. 28公路上行驶的汽车急刹车时的行驶路程s(m)s(m)与时间t(s)t(s)的函数关系式为s=20t−5t2s=20t-5t^{2}，当遇到紧急情况时，司机急刹车，但由于惯性汽车至多要滑行()(\quad) A. 1010米 B. 2020米…
  29. 29The line of symmetry of a quadratic curve is x=−2x = -2 and the curve lies above the xx-axis for all xx. Given that the…
  30. 30A tank is firing rounds during a target practice. The height, hh metres, of a projectile above the ground during flight…
  31. 31The diagram shows part of the curve y=(9−x)(x−3)y=(9-x)(x-3) and the line y=k−3y=k-3, where k>3k>3. The line through the maximum point…
  32. 32Express y=4x2+24x+49y = 4x^2 + 24x + 49 in the form a(x+b)2+ca(x+b)^2 + c. Hence, state the maximum value of 1y\frac{1}{y} and the corres…
  33. 33A circle CC has equation x2+y2+6x−4y=12x^2 + y^2 + 6x - 4y = 12. Find the centre and the radius of CC. The points P(−8,2)P(-8, 2) and Q(1,−1)Q(1, -1)…
  34. 34The equation of a curve is y=2x2−6x+9y = 2x^2 - 6x + 9. By expressing 2x2−6x+92x^2 - 6x + 9 in the form a(x+b)2+ca(x+b)^2 + c, where aa, bb…
  35. 35Express the equations [equations missing from original image] and [equations missing from original image] in the form of…
  36. 36等差数列{an}\{a_{n}\}的前nn项和为SnS_{n}，已知a1=13a_{1}=13，S3=S11S_{3}=S_{11}，当SnS_{n}最大时，nn的值是()() A. 55 B. 66 C. 77…
  37. 37设 f(x)=2x2−10x+6f(x) = 2x^2 - 10x + 6，x≤kx \leq k。 a 求使 f−1(x)f^{-1}(x) 存在的最大 kk 值。 b 对此 kk 值：i 求 f−1(x)f^{-1}(x)。ii 写出 f−1(x)f^{-1}(x)…
  38. 38Express 5−4x−2x25 - 4x - 2x^{2} in the form c−a(x+b)2c - a (x + b)^{2}, where aa, bb and cc are constants and a>0a > 0. Sketch the…
  39. 39Express −4x2+12x+2-4x^2 + 12x + 2 in the form a(x+b)2+ca(x + b)^2 + c and state the coordinates of the turning point of the curve y=−4x2+12x+2y = -4x^2 + 12x + 2…
  40. 40Answer ALL the questions Express 3x2−8x−33x^2 - 8x - 3 in the form a(x+b)2+ca(x + b)^2 + c, where a,ba, b and cc are constants to be…
  41. 41已知实数mm，nn满足m−n2=2m-n^{2}=2，则代数式m2+2n2+4m−3m^{2}+2n^{2}+4m-3的最小值等于()() A. 99 B. 66 C. −8-8 D. −16-16
  42. 42不论xx、yy为何有理数时，x2+y2−12x+4y+40x^{2}+y^{2}-12x+4y+40的值均为()() A. 正数 B. 零 C. 负数 D. 非负数
  43. 43Factorise completely x4−y4x^4 - y^4. Answer ... Given that x2−2xy=1−y2x^2 - 2xy = 1 - y^2, express yy in terms of xx. Answer ... o…
  44. 44Express 2x2−3x−112x^2 - 3x - 11 in the form a(x−h)2+ka(x - h)^2 + k, where aa, hh and kk are constants. Hence, state the coordinate…
  45. 45Find the range of values of xx which satisfies x2+2x+2x2−5x+6≥0.\frac{x^2 + 2x + 2}{x^2 - 5x + 6} \ge 0. Show that the line y=x+ky = x + k…
  46. 46Express −2x2+3x+2-2x^{2} + 3x + 2 in the form a(x+b)2+ca(x + b)^{2} + c where aa, bb and cc are constants. Use your answer from par…
  47. 47已知3x+y=103x{+}y{=}10，则x2+y2x^{2}{+}y^{2}的最小值为()({ }) A. 110\dfrac{1}{10} B. 1010 C. 11 D. 100100
  48. 48Express y=2x2−4x+7y = 2x^2 - 4x + 7 in the form a(x−b)2+ca(x - b)^2 + c, where aa, bb and cc are constants. Hence, explain why the c…
  49. 49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) …
  50. 50在直角坐标系xOyxOy中，直线ll的参数方程为{y=−5+12tx=−32t(t \begin{cases} \overset{x=- \dfrac { \sqrt {3}}{2}t}{y=-5+ \dfrac {1}{2}t}\end{cases}(t为…
  51. 51A stone is thrown vertically upwards such that its height, hh metres from the ground at time tt seconds after being th…
  52. 52A curve has the equation y=4x2−24x+30y=4x^2-24x+30. A line of gradient mm passes through the point (0,−10)(0,-10). Express 4x2−24x+304x^2-24x+30…
  53. 53Explain why 3x2+15x+203x^2+15x+20 cannot be smaller than 1.
  54. 54The height of a coin from the ground, hh meters, after it has been flipped in the air for tt seconds, can be represent…
  55. 55Answer ALL the questions Express 3x2−8x−33x^2 - 8x - 3 in the form a(x+b)2+ca(x + b)^2 + c, where aa, bb and cc are constants…
  56. 56某商品的进价为每件3030元，现在的售价为每件4040元，每星期可卖出150150件.市场调查反映：如果每件的售价每涨11元((售价每件不能高于4545元))，那么每星期少卖1010件.设每件涨价xx…
  57. 57Explain by completing the square that x2−2x+3x^2 - 2x + 3 is always positive for all real values of xx. Hence, find the rang…
  58. 58已知等差数列{an}\{a _{n} \}的前nn项和为Sn.S _{n} .若a1=9a _{1} =9，公差d=−2d=-2，则SnS _{n}的最大值为___．