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Flashcards · Maths

Functions, Sequences and Rates of Change

80 cards from 5 lessons

  1. What is a function?

    A rule that gives exactly one output for each input.

  2. What does \(f(4)\) mean?

    The output when the input is 4.

  3. How do you work out \(f(a)\)?

    Replace \(x\) with \(a\) in the rule.

  4. What does \(fg(x)\) mean?

    \(f(g(x))\), where \(g\) is applied first.

  5. Which function is applied first in \(gf(x)\)?

    \(f\).

  6. Are \(fg(x)\) and \(gf(x)\) always equal?

    No.

  7. What does \(ff(x)\) mean?

    \(f(f(x))\), applying \(f\) twice.

  8. What is \(f^{-1}(x)\)?

    The inverse function of \(f\).

  9. Is \(f^{-1}(x)\) the same as \(\dfrac{1}{f(x)}\)?

    No, it is the inverse function, not a reciprocal.

  10. What are the steps to find an inverse?

    Write \(y = f(x)\), make \(x\) the subject, then swap \(y\) and \(x\).

  11. What is the inverse of \(f(x) = x + 5\)?

    \(f^{-1}(x) = x - 5\).

  12. What is the inverse of \(f(x) = 3x\)?

    \(f^{-1}(x) = \dfrac{x}{3}\).

  13. What is \(f(f^{-1}(x))\)?

    \(x\).

  14. If \(f(x) = 2x + 1\), what is \(f(3)\)?

    7.

  15. How do you check an inverse?

    Try a number through the function and then the inverse.

  16. Which tier are functions?

    Higher tier on all boards.

  17. What is a geometric sequence?

    A sequence where each term is found by multiplying by the same number.

  18. What is the common ratio?

    The number you multiply by each time.

  19. How do you find the common ratio?

    Divide a term by the one before it.

  20. What is the common ratio of \(80, 40, 20, 10\)?

    \(\dfrac{1}{2}\).

  21. What happens if the ratio is negative?

    The terms alternate between positive and negative.

  22. What is the \(n\)th term of a geometric sequence?

    \(a \times r^{n-1}\).

  23. What is the 6th term of \(2, 4, 8, \ldots\)?

    64.

  24. What is a Fibonacci-type sequence?

    Each term is the sum of the two terms before it.

  25. What is the next term of \(1, 1, 2, 3, 5, 8\)?

    13.

  26. What are the square numbers?

    \(1, 4, 9, 16, 25, \ldots\).

  27. What are the cube numbers?

    \(1, 8, 27, 64, \ldots\).

  28. What are the triangular numbers?

    \(1, 3, 6, 10, 15, \ldots\).

  29. If the 2nd term is 6 and the 5th is 48, what is \(r^3\)?

    8.

  30. What is the ratio of \(2, 2\sqrt{3}, 6\)?

    \(\sqrt{3}\).

  31. How do you tell arithmetic from geometric?

    Arithmetic adds a constant, geometric multiplies by a constant.

  32. Which part of this topic is Higher tier?

    Finding the ratio from two terms, and a surd ratio.

  33. What is iteration?

    Repeating a calculation, using each answer as the next input.

  34. What does \(x_{n+1} = f(x_n)\) mean?

    The next value is found by putting the current value into \(f\).

  35. What is \(x_0\)?

    The starting value.

  36. What is the limit of an iteration?

    The value the terms get closer and closer to.

  37. What is true at the limit?

    \(x_{n+1} = x_n\), so \(a = f(a)\).

  38. How do you find the limit exactly?

    Solve \(a = f(a)\).

  39. What does a change of sign tell you?

    A root lies between the two values, if the function is continuous.

  40. What is the change of sign test for \(x^3 + x - 3\) between 1 and 2?

    \(f(1) = -1\) and \(f(2) = 7\), so there is a root.

  41. Why is the root of \(x^3 + x - 3\) rounded to 1.2?

    \(f(1.25) > 0\) and \(f(1.2) < 0\), so it is below 1.25.

  42. How do you get \(x = 3 - \dfrac{2}{x}\) from \(x^2 - 3x + 2 = 0\)?

    Divide by \(x\) and rearrange.

  43. What should you do with the answer from each step?

    Use it as the next input.

  44. What is \(x_1\) if \(x_{n+1} = 3 - \dfrac{2}{x_n}\) and \(x_0 = 4\)?

    2.5.

  45. Which roots does the iteration \(x_{n+1} = 3 - \dfrac{2}{x_n}\) find?

    The roots of \(x^2 - 3x + 2 = 0\), 1 or 2.

  46. Why do you need a continuous function for a change of sign?

    There must be no gaps where the graph jumps over zero.

  47. Which tier is iteration?

    Higher tier on all boards.

  48. What should you show for each iteration?

    Each value, so that the working is clear.

  49. What is a tangent?

    A straight line that touches a curve at one point.

  50. How do you find the gradient of a curve at a point?

    Draw a tangent and find its gradient.

  51. What is a chord?

    A straight line joining two points on a curve.

  52. What does the gradient of a chord give?

    The average rate of change.

  53. What does the gradient of a distance-time graph represent?

    Speed.

  54. What does the gradient of a velocity-time graph represent?

    Acceleration.

  55. What does the area under a velocity-time graph represent?

    Distance travelled.

  56. What is the area of a trapezium?

    \(\dfrac{1}{2}(a + b)h\).

  57. How do you estimate the area under a curve?

    Split it into strips and add the areas of the trapezia.

  58. What is the unit of acceleration?

    m/s\(^2\).

  59. When is a trapezium estimate an underestimate?

    When the curve bends downwards.

  60. When is a trapezium estimate an overestimate?

    When the curve bends upwards.

  61. What is the average rate of change of \(y = x^2\) from \(x = 1\) to \(x = 4\)?

    5.

  62. What makes the estimate more accurate?

    More, narrower strips.

  63. Why should the two points on a tangent be far apart?

    To get a more accurate gradient.

  64. Which tier is this topic?

    Higher tier on all boards.

  65. What does \(y = f(x) + a\) do to the graph?

    Translates it up by \(a\).

  66. What does \(y = f(x + a)\) do to the graph?

    Translates it left by \(a\).

  67. What does \(y = f(x - a)\) do to the graph?

    Translates it right by \(a\).

  68. What does \(y = -f(x)\) do to the graph?

    Reflects it in the \(x\)-axis.

  69. What does \(y = f(-x)\) do to the graph?

    Reflects it in the \(y\)-axis.

  70. Where does \((3, 5)\) move to under \(y = f(x) - 4\)?

    \((3, 1)\).

  71. Where does \((3, 5)\) move to under \(y = f(x - 2)\)?

    \((5, 5)\).

  72. Where does \((3, 5)\) move to under \(y = f(-x)\)?

    \((-3, 5)\).

  73. Where does \((3, 5)\) move to under \(y = -f(x)\)?

    \((3, -5)\).

  74. What is the equation of \(y = x^2\) moved 3 units right?

    \(y = (x - 3)^2\).

  75. What is the equation of \(y = x^2\) moved 2 units up?

    \(y = x^2 + 2\).

  76. What is \(\sin(x + 90^\circ)\) equal to?

    \(\cos x\).

  77. What is the maximum of \(y = \sin x + 1\)?

    2.

  78. Which way does a minus outside the bracket reflect?

    In the \(x\)-axis.

  79. How do you sketch a transformation?

    Follow a key point, such as a turning point, and redraw.

  80. Which tier are graph transformations?

    Higher tier on all boards.