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Form 6 Functions and Graphs Synthesis Practice

Form 6 · 60 practice questions with instant marking and worked solutions.

Functions and Graphs Synthesis is an important part of the Form 6 mathematics curriculum. This topic develops function inputs and outputs, intercepts, gradients and transformations. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Functions and Graphs Synthesis has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Functions and Graphs Synthesis; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 8 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Functions and Graphs Synthesis set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Functions and Graphs Synthesis should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    In Functions and Graphs Synthesis, calculate $y=2x+4,\ x=4$. Choose the correct answer.
  2. 2 Basic
    In Functions and Graphs Synthesis, calculate $y=3x+5,\ x=5$. Give the numerical value only.

  3. 3 Basic
    In Functions and Graphs Synthesis, calculate $y=4x+4,\ x=6$. Show the essential working and give the numerical value.

  4. 4 Basic
    In Functions and Graphs Synthesis, the value of $y=5x+5,\ x=7$ is 41. Is this statement true?
  5. 5 Basic
    It is given that $f(x)=-x^{2}-9$. Find the value of $f(0)$. Choose the correct answer.
  6. 6 Basic
    Consider the quadratic function $y=x^{2}-2x+4$. Find the minimum value of $y$. Give the answer only.

  7. 7 Basic
    Consider the parabola $y=4x^{2}+4x+6$. Find the value of $h$ in its axis of symmetry $x=h$ (as a fraction in lowest terms). Show the essential working and give the answer.

  8. 8 Basic
    The vertex of $y=f(x)$ is $(2,-2)$. Consider $y=f(x-1)-2$. The $y$-coordinate of the new vertex is 0. Is this true?
  9. 9 Basic
    It is given that $f(x)=3x^{2}-6x+7$. Find the value of $f(2)$. Choose the correct answer.
  10. 10 Basic
    Consider the quadratic function $y=x^{2}+12x+28$. Find the minimum value of $y$. Give the answer only.

  11. 11 Basic
    Consider the parabola $y=2x^{2}+6x-7$. Find the value of $h$ in its axis of symmetry $x=h$ (as a fraction in lowest terms). Show the essential working and give the answer.

  12. 12 Basic
    The vertex of $y=f(x)$ is $(-1,-6)$. Consider $y=f(x+4)+1$. The $y$-coordinate of the new vertex is -7. Is this true?
  13. 13 Standard
    In Functions and Graphs Synthesis, calculate $y=1x+7,\ x=2$. Choose the correct answer.
  14. 14 Standard
    In Functions and Graphs Synthesis, calculate $y=2x+8,\ x=3$. Give the numerical value only.

  15. 15 Standard
    In Functions and Graphs Synthesis, calculate $y=3x+7,\ x=4$. Show the essential working and give the numerical value.

  16. 16 Standard
    In Functions and Graphs Synthesis, the value of $y=4x+8,\ x=5$ is 28. Is this statement true?
  17. 17 Standard
    The point $(-6,2)$ lies on $y=f(x)$, and $Q$ is the corresponding point on $y=f(-x)$. Find the $x$-coordinate of $Q$. Choose the correct answer.
  18. 18 Standard
    A line passes through $A(4,2)$ and $B(-5,9)$. Find the gradient of $AB$ (as a fraction in lowest terms). Give the answer only.

  19. 19 Standard
    The parabola $y=x^{2}+x-12$ cuts the $x$-axis at two points. Find the distance between the two $x$-intercepts. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    $f(x)=x^{2}-8x+19$. The graph of $y=f(x)$ is translated 4 units to the right and then 1 unit up to give $y=g(x)$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the vertex of $y=f(x)$.
    2. (b) Find the $x$-coordinate of the vertex of $y=g(x)$.
    3. (c) Find the $y$-coordinate of the vertex of $y=g(x)$.
    4. (d) Find the $y$-intercept of $y=g(x)$.
  21. 21 Standard
    The point $(2,-3)$ lies on $y=f(x)$, and $Q$ is the corresponding point on $y=f(-x)$. Find the $x$-coordinate of $Q$. Choose the correct answer.
  22. 22 Standard
    A line passes through $A(-1,1)$ and $B(0,3)$. Find the gradient of $AB$. Give the answer only.

  23. 23 Standard
    The parabola $y=x^{2}+3x-18$ cuts the $x$-axis at two points. Find the distance between the two $x$-intercepts. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    $f(x)=x^{2}-8x+15$. The graph of $y=f(x)$ is translated 3 units to the right and then 4 units down to give $y=g(x)$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the vertex of $y=f(x)$.
    2. (b) Find the $x$-coordinate of the vertex of $y=g(x)$.
    3. (c) Find the $y$-coordinate of the vertex of $y=g(x)$.
    4. (d) Find the $y$-intercept of $y=g(x)$.
  25. 25 Advanced
    In Functions and Graphs Synthesis, calculate $y=5x+10,\ x=6$. Choose the correct answer.
  26. 26 Advanced
    In Functions and Graphs Synthesis, calculate $y=1x+11,\ x=7$. Give the numerical value only.

  27. 27 Advanced
    In Functions and Graphs Synthesis, calculate $y=2x+10,\ x=2$. Show the essential working and give the numerical value.

  28. 28 Advanced
    In Functions and Graphs Synthesis, the value of $y=3x+11,\ x=3$ is 21. Is this statement true?
  29. 29 Advanced
    Let $f(x)=2x$ and $g(x)=-x^{2}-2$. Find the value of $g(f(0))$. Choose the correct answer.
  30. 30 Advanced
    It is given that $f(x)=3x^{2}-4x-5$. Find the value of $f(-3)$. Give the answer only.

  31. 31 Advanced
    Consider the parabola $y=4x^{2}+14x+3$. The value of $h$ in its axis of symmetry $x=h$ is -7/4. Is this true?
  32. 32 Advanced DSE long question
    The line $L:y=3x+4$ meets the parabola $\Gamma:y=x^{2}+6x+4$ at $A$ and $B$ ($A$ to the left of $B$). Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of $B$.
    2. (b) Find the $y$-coordinate of $B$.
    3. (c) Find the sum of the $y$-coordinates of $A$ and $B$.
  33. 33 Advanced
    Let $f(x)=3x+1$ and $g(x)=2x^{2}+3$. Find the value of $g(f(0))$. Choose the correct answer.
  34. 34 Advanced
    It is given that $f(x)=-x^{2}-6x+1$. Find the value of $f(2)$. Give the answer only.

  35. 35 Advanced
    Consider the quadratic function $y=2x^{2}-16x+37$. The minimum value of $y$ is 5. Is this true?
  36. 36 Advanced DSE long question
    The line $L:y=-2x-4$ meets the parabola $\Gamma:y=x^{2}-6x-9$ at $A$ and $B$ ($A$ to the left of $B$). Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of $B$.
    2. (b) Find the $y$-coordinate of $B$.
    3. (c) Find the sum of the $y$-coordinates of $A$ and $B$.
  37. 37 Challenging
    In Functions and Graphs Synthesis, calculate $y=4x+13,\ x=4$. Choose the correct answer.
  38. 38 Challenging
    In Functions and Graphs Synthesis, calculate $y=5x+14,\ x=5$. Give the numerical value only.

  39. 39 Challenging
    In Functions and Graphs Synthesis, calculate $y=1x+13,\ x=6$. Show the essential working and give the numerical value.

  40. 40 Challenging
    In Functions and Graphs Synthesis, the value of $y=2x+14,\ x=7$ is 28. Is this statement true?
  41. 41 Challenging
    Consider the parabola $y=x^{2}+7x$. Find the value of $h$ in its axis of symmetry $x=h$ (as a fraction in lowest terms). Choose the correct answer.
  42. 42 Challenging
    The vertex of $y=f(x)$ is $(5,-1)$. Consider $y=f(x-2)+4$. Find the $y$-coordinate of the new vertex. Show the essential working and give the answer.

  43. 43 Challenging
    The point $(3,4)$ lies on $y=f(x)$, and $Q$ is the corresponding point on $y=f(3x)$. The $x$-coordinate of $Q$ is 3. Is this true?
  44. 44 Challenging DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-5$ and $2$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(-3)$.
    4. (d) Find the minimum value of $f(x)$ (as a fraction in lowest terms).
  45. 45 Challenging
    Consider the parabola $y=x^{2}-15x-5$. Find the value of $h$ in its axis of symmetry $x=h$ (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    The vertex of $y=f(x)$ is $(-4,5)$. Consider $y=f(x-1)-5$. Find the $y$-coordinate of the new vertex. Show the essential working and give the answer.

  47. 47 Challenging
    The point $(-5,-4)$ lies on $y=f(x)$, and $Q$ is the corresponding point on $y=f(4x)$. The $x$-coordinate of $Q$ is -5. Is this true?
  48. 48 Challenging DSE long question
    It is given that $f(x)=x^2+bx+c$, where the roots of $f(x)=0$ are $-4$ and $1$. Answer each part; give the answer only.
    1. (a) Find $b$.
    2. (b) Find $c$.
    3. (c) Find $f(1)$.
    4. (d) Find the minimum value of $f(x)$ (as a fraction in lowest terms).
  49. 49 Elite
    In Functions and Graphs Synthesis, calculate $y=3x+16,\ x=2$. Choose the correct answer.
  50. 50 Elite
    In Functions and Graphs Synthesis, calculate $y=4x+17,\ x=3$. Give the numerical value only.

  51. 51 Elite
    In Functions and Graphs Synthesis, calculate $y=5x+16,\ x=4$. Show the essential working and give the numerical value.

  52. 52 Elite
    In Functions and Graphs Synthesis, the value of $y=1x+17,\ x=5$ is 23. Is this statement true?
  53. 53 Elite
    A line passes through $A(2,0)$ and $B(-2,-2)$. Find the gradient of $AB$ (as a fraction in lowest terms). Give the answer only.

  54. 54 Elite
    The parabola $y=x^{2}-5x-14$ cuts the $x$-axis at two points. Find the distance between the two $x$-intercepts. Show the essential working and give the answer.

  55. 55 Elite
    Let $f(x)=3x+6$ and $g(x)=-x^{2}+1$. The value of $g(f(-2))$ is 1. Is this true?
  56. 56 Elite DSE long question
    $f(x)=x^{2}-4x+9$. The graph of $y=f(x)$ is translated 4 units to the right and then 4 units up to give $y=g(x)$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the vertex of $y=f(x)$.
    2. (b) Find the $x$-coordinate of the vertex of $y=g(x)$.
    3. (c) Find the $y$-coordinate of the vertex of $y=g(x)$.
    4. (d) Find the $y$-intercept of $y=g(x)$.
  57. 57 Elite
    A line passes through $A(-1,-3)$ and $B(-5,7)$. Find the gradient of $AB$ (as a fraction in lowest terms). Give the answer only.

  58. 58 Elite
    The parabola $y=x^{2}-12x+20$ cuts the $x$-axis at two points. Find the distance between the two $x$-intercepts. Show the essential working and give the answer.

  59. 59 Elite
    Let $f(x)=3x+2$ and $g(x)=x^{2}+3$. The value of $g(f(-2))$ is 19. Is this true?
  60. 60 Elite DSE long question
    $f(x)=x^{2}+8x+21$. The graph of $y=f(x)$ is translated 2 units to the right and then 4 units down to give $y=g(x)$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the vertex of $y=f(x)$.
    2. (b) Find the $x$-coordinate of the vertex of $y=g(x)$.
    3. (c) Find the $y$-coordinate of the vertex of $y=g(x)$.
    4. (d) Find the $y$-intercept of $y=g(x)$.

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Functions and Graphs Synthesis

This topic needs particular care with function inputs and outputs, intercepts, gradients and transformations. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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