Secondary 2 Mathematics Classroom | Chapter 6: Quadratic Expressions, Equations, Functions and Graphs | G2/G3

SECONDARY 2 MATHEMATICS CLASSROOM · CHAPTER 6 · QUADRATIC EXPRESSIONS · EQUATIONS · FUNCTIONS · GRAPHS · G2/G3

Quadratic Expressions, Equations, Functions and Graphs: When a Straight Line Starts to Bend

A quadratic expression can be expanded or factorised. A quadratic equation asks for roots. A quadratic function creates a curved graph. The same algebraic structure appears in three different mathematical jobs.

Chapter 5 studied straight-line relationships and simultaneous linear equations. Chapter 6 introduces second-degree structure. For G3 Secondary 2, quadratic equations by factorisation and quadratic functions/graphs form a major development. For G2, the shared Secondary 2 foundation is the quadratic-expression and factorisation work; quadratic equations, functions and graphs in this classroom are clearly marked as later-course or bridge material rather than presented as identical G2 Secondary 2 core.

Classroom rule: identify the mathematical object → read the level boundary → choose the useful form → factorise or build the table → connect roots to x-intercepts → use symmetry to organise the graph → interpret only inside the permitted domain.

Level boundary. G3 students should use the full classroom as current Secondary 2 instruction. G2 students should master the shared expression/factorisation foundation and use sections labelled G3 / later-course bridge only when their school’s current sequence has introduced them. The aim is to keep progression honest while preserving one connected mathematical map.

Official reference: MOE G2 and G3 Mathematics Syllabuses.

Navigate: retrieval · quadratic expressions · factorisation · quadratic equations · zero-product property · quadratic functions · shape and leading coefficient · symmetry and turning points · intercepts and roots · tables and sketches · modelling · misconception clinic · guided practice · assessment transfer · exit ticket.

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Featured Answer: What Makes an Expression Quadratic?

A quadratic expression in one variable has highest power 2, for example x²+5x+6 or 3x²−4x+1. When it is placed inside an equation such as x²+5x+6=0, the task changes from manipulating an expression to finding values of x that make the equality true.

Expression: manipulate the form. Equation: find roots. Function: study input-output behaviour and graph features.

How to Use This Classroom

1. Read the level marker before using a section.
2. Distinguish expression, equation and function before doing any algebra.
3. Check for a common factor before quadratic factorisation.
4. Use factorisation only when the structure supports it.
5. For equations, rearrange to zero before using the zero-product property.
6. For graphs, predict opening direction before calculating points.
7. Use roots to locate x-intercepts.
8. Use the midpoint of two roots to locate the symmetry line.
9. Use substitution to find the turning-point y-coordinate.
10. Return every algebraic answer to the original context and domain.

1. Retrieval: Expansion and Factorisation Are Reverse Views

(x+2)(x+3)=x²+5x+6. Therefore x²+5x+6=(x+2)(x+3).

2. The x² Term Comes From Multiplying Variable Terms

In (x+2)(x+3), the product x×x creates x². This is what makes the expanded expression quadratic rather than linear.

3. An Expression Is Not Yet an Equation

x²+5x+6 can be factorised. x²+5x+6=0 can be solved.

4. Factorisation Is Not Yet Solving

(x+2)(x+3) is a factorised expression. The roots −2 and −3 appear only when the product is set equal to zero.

5. Quick Retrieval Diagnostic

1. Expand (x+4)(x+2).
2. Factorise x²+7x+12.
3. Factorise 2x²+7x+3.
4. State the difference between factorising x²−5x+6 and solving x²−5x+6=0.

Answers

x²+6x+8. (x+3)(x+4). (2x+1)(x+3). Factorisation gives an equivalent product; solving finds the x-values making the equation true.

6. Shared G2/G3 Core: Read the Quadratic Structure

For ax²+bx+c, the coefficient a belongs to x², b belongs to x, and c is the constant. The expression is quadratic when a is non-zero.

7. Teacher Model 1: Identify Coefficients

For 3x²−7x+5:

• a=3;
• b=−7;
• c=5.

8. The Sign Belongs to the Coefficient

In 3x²−7x+5, the linear coefficient is −7, not 7.

9. Equivalent Forms Reveal Different Features

x²−5x+6 and (x−2)(x−3) are equivalent. Expanded form shows coefficients; factor form shows zeros of the product.

10. Teacher Model 2: Expand Before Comparing

(2x−1)(x+4)=2x²+8x−x−4=2x²+7x−4.

11. Do Not Collect x² and x as Like Terms

2x²+7x−4 is already collected by powers. x² and x are different algebraic units.

12. Shared G2/G3 Core: Common Factor First

Before looking for two brackets, check whether every term shares a factor.

13. Teacher Model 3: Common Factor

Factorise 3x²−15x.

3x(x−5).

14. Simple Monic Quadratics Use Sum and Product

For x²+px+q, find numbers whose sum is p and product is q.

15. Teacher Model 4: Positive Product

x²+9x+20=(x+4)(x+5).

16. Teacher Model 5: Negative Product

x²+x−12=(x+4)(x−3).

17. Positive Product and Negative Sum Means Both Signs Are Negative

x²−11x+24=(x−3)(x−8).

18. Teacher Model 6: Leading Coefficient Greater Than One

Factorise 2x²+7x+3.

2x²+6x+x+3=2x(x+3)+1(x+3).

(2x+1)(x+3).

19. Re-Expansion Is the Structural Check

(2x+1)(x+3)=2x²+7x+3.

20. Identity Patterns Can Be Read in Reverse

a²−b²=(a−b)(a+b)
a²−2ab+b²=(a−b)²
a²+2ab+b²=(a+b)²

21. Teacher Model 7: Difference of Squares

9x²−16=(3x−4)(3x+4).

22. Teacher Model 8: Perfect Square

4x²−12x+9=(2x−3)².

Your Turn 1 — Shared G2/G3 Core

1. Factorise x²+8x+15.
2. Factorise x²−2x−15.
3. Factorise 3x²+10x+3.
4. Factorise 16x²−25.
5. Factorise 9x²−24x+16.

Answers

(x+3)(x+5). (x−5)(x+3). (3x+1)(x+3). (4x−5)(4x+5). (3x−4)².

23. G3 Secondary 2 / G2 Later-Course Bridge: A Quadratic Equation Asks for Roots

x²+5x+6=0 asks which x-values make the quadratic expression equal to zero.

24. The Zero-Product Property Connects Factors to Roots

If AB=0, then at least one factor is zero. Therefore (x+2)(x+3)=0 gives x=−2 or x=−3.

25. The Product Must Equal Zero

If (x−4)(x+7)=12, you may not set either factor to zero. First form an equivalent equation with zero on one side.

26. Teacher Model 9: Already Factorised Equation

(2x+3)(x−5)=0.

2x+3=0 or x−5=0.

x=−3/2 or x=5.

27. Factorise Completely Before Solving

x²−11x+24=0 becomes (x−3)(x−8)=0.

28. Teacher Model 10: Monic Quadratic Equation

x²−11x+24=0.

x=3 or x=8.

29. Teacher Model 11: Non-Unit Leading Coefficient

2x²+7x+3=0.

(2x+1)(x+3)=0.

x=−1/2 or x=−3.

30. Common-Factor Equations Can Have Zero as a Root

3x²−15x=0 becomes 3x(x−5)=0, so x=0 or x=5.

31. Do Not Divide Away a Possible Zero Root

Dividing x(x−5)=0 by x assumes x≠0 and loses a valid solution.

32. Rearrange to Zero Before Factorising

x²=5x+14 becomes x²−5x−14=0.

33. Teacher Model 12: Rearrange Then Solve

x²=5x+14.

x²−5x−14=0=(x−7)(x+2).

x=7 or x=−2.

34. Teacher Model 13: Product Equals a Non-Zero Number

x(x+4)=12.

x²+4x−12=0=(x+6)(x−2).

x=−6 or x=2.

35. A Repeated Factor Gives One Repeated Root

(x−3)²=0 gives x=3. The root appears twice in the factorisation but has one numerical value.

Your Turn 2 — G3 / Later-Course Bridge

1. Solve x²+7x+12=0.
2. Solve x²−x−20=0.
3. Solve 2x²+5x+2=0.
4. Solve 4x²−20x=0.
5. Solve x²=6x+16.

Answers

x=−3 or −4. x=5 or −4. x=−1/2 or −2. x=0 or 5. x=8 or −2.

36. G3 Secondary 2 / G2 Later-Course Bridge: A Quadratic Function Assigns an Output to Each Input

Consider y=x²−4x+3. Every chosen x produces one y-value.

37. The Graph Is the Set of All Coordinate Pairs Satisfying the Rule

A table lists selected points. The equation describes the rule exactly. The graph displays the overall behaviour.

38. Teacher Model 14: Build a Table

For y=x²−4x+3:

x	−1	0	1	2	3	4	5
y	8	3	0	−1	0	3	8

39. Negative Inputs Need Brackets During Substitution

At x=−1, x²=(−1)²=1 and −4x=+4, giving y=8.

40. Quadratic First Differences Are Not Constant

Unlike a straight line, a quadratic does not have one constant gradient across the whole graph.

41. Equal x-Steps Produce Constant Second Differences

For y=x², first differences 1,3,5,7 have second differences 2,2,2. This is a useful pattern check for equally spaced inputs.

42. The Sign of the x² Coefficient Controls Opening Direction

For y=ax²+bx+c:

• a>0 → parabola opens upward and has a minimum;
• a<0 → parabola opens downward and has a maximum.

43. Teacher Model 15: Predict Before Plotting

y=−2x²+8x−5 opens downward because the leading coefficient is negative.

44. The Magnitude of a Also Changes the Apparent Width

y=2x² grows faster in magnitude than y=½x² as |x| increases, so the graph appears narrower on equally scaled axes.

45. b and c Affect Position and Intercepts

The leading coefficient controls opening direction, but it does not determine the entire graph.

46. Every Parabola Has a Vertical Line of Symmetry

Points equally far left and right of the symmetry line have equal y-values.

47. Two Roots Reveal the Symmetry Line by Their Midpoint

If the roots are r₁ and r₂, the symmetry line lies halfway between them.

48. Teacher Model 16: Roots to Symmetry

y=(x−1)(x−5) has roots 1 and 5.

Midpoint=(1+5)/2=3.

Symmetry line: x=3.

49. Substitute the Symmetry x-Value to Find the Turning Point

For y=(x−1)(x−5), y at x=3 is 2(−2)=−4.

Turning point: (3,−4).

50. Positive Leading Coefficient Makes That Turning Point a Minimum

Nearby y-values are higher than −4.

51. Equal Outputs Can Reveal Symmetry When the Quadratic Structure Is Known

If corresponding points have y=7 at x=−1 and x=5, the symmetry line is halfway at x=2.

52. Do Not Infer Symmetry From Two Equal Values Without the Quadratic Context

The conclusion depends on knowing these are corresponding points on the same parabola.

53. The y-Intercept Occurs at x=0

For y=ax²+bx+c, the y-intercept is (0,c).

54. The x-Intercepts Are the Roots of ax²+bx+c=0

Setting y=0 turns the function into a quadratic equation.

55. Teacher Model 17: Factor Form Shows Roots Directly

y=x²−6x+8=(x−2)(x−4).

x-intercepts: (2,0) and (4,0).

56. Their Midpoint Gives the Symmetry Line

x=(2+4)/2=3.

57. Substitute x=3 to Find the Turning Point

y=9−18+8=−1, so turning point is (3,−1).

58. Expanded Form Shows the y-Intercept

At x=0, y=8, so the y-intercept is (0,8).

59. A Quadratic Can Cross Twice, Touch Once or Miss the x-Axis

• two crossings → two distinct real roots;
• one touch → one repeated real root;
• no meeting → no real roots.

60. Teacher Model 18: Repeated Root

y=(x−3)² touches the x-axis at (3,0). The equation (x−3)²=0 has repeated root x=3.

61. Teacher Model 19: No Real x-Intercept

y=x²+4 is always at least 4 for real x, so it never meets the x-axis.

62. A Controlled Sketch Combines Structure and Calculated Points

Identify opening direction, roots if available, symmetry line, turning point and y-intercept. Add symmetric points as needed, then draw a smooth parabola.

63. Do Not Join Quadratic Points With Straight Segments

The plotted points sample a smooth curved relationship.

64. Teacher Model 20: Sketch y=x²+2x−3

Factorise: (x+3)(x−1).

• roots: −3 and 1;
• symmetry line: x=−1;
• turning point: (−1,−4);
• y-intercept: (0,−3);
• opens upward.

65. Symmetry Can Check a Table

For symmetry line x=−1, outputs at x=−2 and x=0 should match.

66. Do Not Use Symmetry to Hide an Incorrect Axis

Establish the axis from valid information first.

Your Turn 3 — G3 / Later-Course Bridge

1. For y=x²−4x+3, state the roots, symmetry line, turning point and y-intercept.
2. For y=(x−2)(x+4), find the symmetry line and turning point.
3. Does y=−x²+6x−5 have a maximum or minimum?
4. Explain why y=(x−4)² has one repeated root.

Answers

Roots 1,3; symmetry x=2; turning point (2,−1); y-intercept (0,3). Symmetry x=−1; turning point (−1,−9). Maximum. The graph touches the x-axis at x=4 and the factor x−4 is repeated.

67. Quadratic Models Need a Meaningful Domain

A formula can be mathematically defined for more x-values than the real situation allows.

68. Teacher Model 21: Rectangle With Fixed Sum of Sides

A rectangle has side lengths x and 10−x.

Area A=x(10−x)=−x²+10x.

Physical domain: 0<x<10.

69. Roots at 0 and 10 Give the Symmetry Line x=5

At x=5, A=25.

Maximum area: 25 square units.

70. The Turning Point Must Be Interpreted Using the Variables

Here x=5 means a side length of 5, and A=25 means the area. The same coordinate pair in another model could mean something entirely different.

71. Negative Area Outputs Outside the Physical Domain Do Not Represent the Rectangle

They belong to the unrestricted algebraic graph, not to the stated geometry model.

72. Teacher Model 22: Rectangle Equation

A rectangle has width x and length x+4, area 96.

x(x+4)=96.

x²+4x−96=0=(x+12)(x−8).

Roots: −12 and 8. Positive width selects x=8.

Dimensions: 8 by 12.

73. A Negative Root Is Not Automatically Wrong

It is rejected here because width must be positive. In an unrestricted algebraic equation, negative roots can be valid.

74. Teacher Model 23: Two Consecutive Positive Integers

n(n+1)=156.

n²+n−156=0=(n+13)(n−12).

Positive condition selects n=12.

Integers: 12 and 13.

75. Teacher Model 24: Two Valid Roots Can Describe the Same Rectangle

Perimeter 30 and area 54 produce x(15−x)=54.

x²−15x+54=0=(x−6)(x−9).

x=6 or 9. These simply swap the two side labels.

Dimensions: 6 by 9.

76. Context Decides Whether to Keep One Root, Both Roots or Neither

Algebra produces candidates. The original problem supplies admissibility conditions.

77. Misconception Clinic: Factorisation Is the Same as Solving

Repair: factorisation changes form; solving finds values satisfying an equation.

78. Misconception Clinic: Set Factors to Zero When the Product Equals 12

The zero-product property only applies when the product equals zero.

79. Misconception Clinic: Divide Away x in x(x−5)=0

That assumes x≠0 and loses a possible root.

80. Misconception Clinic: Every Quadratic Must Factorise Nicely

Do not invent integer factors when no suitable pair exists.

81. Misconception Clinic: Negative x Gives Negative x²

If x=−3, then x²=(−3)²=9.

82. Misconception Clinic: A Quadratic Graph Has One Constant Gradient

Its rate of change varies with x; that is why the graph curves.

83. Misconception Clinic: Positive a Means Positive y Everywhere

Positive a means the parabola opens upward. The graph can still dip below the x-axis.

84. Misconception Clinic: Roots and Turning Point Are the Same

Roots are x-values where y=0. The turning point is the maximum or minimum point.

85. Misconception Clinic: c Is the x-Intercept

In y=ax²+bx+c, c gives the y-intercept because x=0.

86. Misconception Clinic: Use the Entire Algebraic Graph as the Real-World Domain

The context may restrict x to positive, integer or interval values.

87. Misconception Clinic: Treat G3 Secondary 2 Quadratic Functions as Identical G2 Secondary 2 Core

Repair: follow the current subject-level sequence. G2 students use the shared foundation and move into function/equation sections when their course introduces them.

88. Guided Practice A: Shared G2/G3 Expression Control

1. Expand (x+6)(x+2).
2. Factorise x²+11x+24.
3. Factorise x²−4x−21.
4. Factorise 2x²+9x+4.
5. Factorise 25x²−9.

Solutions

x²+8x+12. (x+3)(x+8). (x−7)(x+3). (2x+1)(x+4). (5x−3)(5x+3).

89. Guided Practice B: G3 / Later-Course Quadratic Equations

1. x²+9x+20=0.
2. x²−7x+12=0.
3. 2x²+7x+3=0.
4. 3x²−15x=0.

Solutions

x=−4,−5. x=3,4. x=−1/2,−3. x=0,5.

90. Guided Practice C: Rearrange to Zero

1. x²=4x+12.
2. x(x+5)=24.
3. x²+3=8x.

Solutions

x²−4x−12=0=(x−6)(x+2), so x=6,−2. x²+5x−24=0=(x+8)(x−3), so x=−8,3. x²−8x+3=0 does not factorise over the integers; do not invent factors.

91. Guided Practice D: Function Tables

1. For y=x²−2x−3, find y at x=−1,0,1,2,3.
2. For y=−x²+4x+1, state opening direction.
3. For y=3x²−5x+7, state y-intercept.

Solutions

0,−3,−4,−3,0. Opens downward. y-intercept (0,7).

92. Guided Practice E: Roots, Symmetry and Turning Points

1. y=(x−2)(x+4): find roots and symmetry line.
2. Find the turning point.
3. y=(x−5)²: state root, symmetry line and turning point.

Solutions

Roots 2,−4; symmetry x=−1. Turning point (−1,−9). Root 5, symmetry x=5, turning point (5,0).

93. Guided Practice F: Sketch Features

For y=x²−4x+3, state every feature needed for a controlled sketch.

Answer

Opens upward; roots 1 and 3; symmetry x=2; turning point (2,−1); y-intercept (0,3). Useful symmetric point: (4,3).

94. Guided Practice G: Modelling and Domain

A rectangle has sides x and 12−x. Find the area function, physical domain and maximum area.

Worked solution

A=x(12−x)=−x²+12x. Domain 0<x<12. Roots 0 and 12 give symmetry x=6. Maximum area A=36 square units.

95. Guided Practice H: Root Interpretation

A rectangle model produces roots −8 and 5 for its width. Which root is admissible?

Answer

Width must be positive, so 5 is admissible and −8 is rejected by the physical context.

96. Challenge Practice: Same Quadratic, Three Jobs

Use x²−6x+8.

1. Factorise it.
2. Solve x²−6x+8=0.
3. For y=x²−6x+8, state roots, symmetry line and turning point.

Answers

(x−2)(x−4). Roots x=2,4. Symmetry x=3; turning point (3,−1).

97. Challenge Practice: One Root Is Repeated

Explain algebraically and graphically what happens for x²−6x+9=0.

Answer

(x−3)²=0 gives repeated root x=3. The graph y=(x−3)² touches the x-axis at (3,0) and turns there.

98. Challenge Practice: Do Not Invent Integer Factors

Can x²+x+1 be factorised into integer linear factors?

Answer

No integer pair multiplies to 1 and adds to 1. Do not create a false factorisation merely because the exercise is near other factorisation questions.

99. Assessment Method: Identify the Job First

• expand → create an equivalent sum;
• factorise → create an equivalent product;
• solve → find roots of an equation;
• graph → study all coordinate pairs of a function;
• model → restrict the mathematics to meaningful values.

100. Assessment Method: State the Level Boundary When It Matters

G2 students should not treat later quadratic-equation/function work as current Secondary 2 core unless their school sequence has introduced it.

101. Assessment Method: Rearrange to Zero Before Zero-Product Reasoning

The factorisation must sit inside an equation with one side equal to zero.

102. Assessment Method: Show Both Roots When Both Are Valid

Do not silently discard a negative or fractional root without a contextual reason.

103. Assessment Method: Predict Graph Shape Before Plotting

The sign of the leading coefficient supplies an immediate check on opening direction.

104. Assessment Method: Use Roots as x-Intercepts

Roots solve y=0 and therefore locate where the graph meets the x-axis.

105. Assessment Method: Use Symmetry as a Check

Corresponding points equally far from the symmetry line should have equal y-values.

106. Assessment Method: State Domain Restrictions in Models

Positive lengths, whole-number counts and stated intervals can remove algebraically possible values.

107. Oral Classroom Check

1. What makes an expression quadratic?
2. What is the difference between factorising and solving?
3. When can the zero-product property be used?
4. Why can x=0 be lost by dividing through by x?
5. What does the sign of the x² coefficient tell you?
6. What is the relationship between roots and x-intercepts?
7. How can two roots locate the symmetry line?
8. What is a turning point?
9. Why can an algebraic root be rejected in a geometry model?
10. What is the G2/G3 boundary in this chapter?

108. Exit Ticket

1. Factorise x²+7x+12.
2. Factorise 2x²+7x+3.
3. G3/bridge: solve x²−5x+6=0.
4. G3/bridge: solve 3x²−12x=0.
5. For y=x²−4x+3, state opening direction and y-intercept.
6. State its roots and symmetry line.
7. Find its turning point.
8. Explain why y=(x−3)² has one repeated root.
9. A width equation gives roots −5 and 8. Which root is physically admissible?
10. Explain why a G2 student may use the first part of this classroom now while treating the function/graph material as bridge work.

Exit-ticket solutions

(x+3)(x+4). (2x+1)(x+3). x=2 or 3. 3x(x−4)=0, so x=0 or 4. Opens upward; y-intercept (0,3). Roots 1 and 3; symmetry x=2. Turning point (2,−1). The factor x−3 occurs twice and the graph touches the x-axis at (3,0). Width must be positive, so 8. Because the subject-level timing differs: shared quadratic-expression/factorisation foundations belong to the G2 route while quadratic equations/functions/graphs are taken when that later course stage is reached.

109. Homework: Retrieval, Variation and Transfer

Layer 1 — Shared G2/G3 Retrieval

• identify a,b,c in three quadratic expressions;
• expand two pairs of brackets;
• factorise two monic quadratics;
• factorise one non-unit quadratic;
• factorise one difference of squares and one perfect square.

Layer 2 — G3 / Later-Course Variation

• solve four factorisable quadratic equations;
• solve two equations that must first be rearranged to zero;
• build two quadratic tables;
• find roots, symmetry and turning points for three functions;
• sketch two parabolas;
• solve one modelling problem with a rejected root.

Layer 3 — Transfer

Choose one factorisable quadratic and present it as an expression, an equation and a function. Explain how the mathematical job changes while the algebraic structure stays the same.

110. The Seven-Day Return Cycle

1. Day 0: factorisation and level-boundary orientation.
2. Day 1: one factorisation; for G3/bridge, one equation and one graph-feature question.
3. Day 3: mixed expression/equation/function classification without headings.
4. Day 7: changed exit ticket with one modelling domain check.

111. A 60-Minute Teaching Lesson

1. 10 minutes: shared expression and factorisation retrieval.
2. 10 minutes: level boundary and mathematical-object distinction.
3. 15 minutes: G3/bridge quadratic equations and zero product.
4. 15 minutes: G3/bridge roots, symmetry and turning points.
5. 5 minutes: modelling/domain interpretation.
6. 5 minutes: exit ticket.

112. A 90-Minute Teaching Lesson

1. 15 minutes: expansion and factorisation diagnostic.
2. 15 minutes: equation versus expression.
3. 20 minutes: factorisation-based quadratic solving.
4. 15 minutes: function tables and graph shape.
5. 15 minutes: symmetry, roots, turning points and sketching.
6. 5 minutes: modelling/domain.
7. 5 minutes: exit ticket and return date.

113. The Full Quadratic-Expression Routine

identify powers → check common factor → recognise a factor pattern → factorise → expand back → confirm exact equivalence.

114. The Full Quadratic-Equation Routine

identify the equation → rearrange to zero → factorise completely → use zero product → list every candidate → substitute/check → apply context restrictions.

115. The Full Quadratic-Graph Routine

read a → predict opening → find roots/intercepts → find symmetry line → calculate turning point → find y-intercept → add symmetric points → draw smooth curve → state domain.

116. Connect Back to Chapter 5

Return to Secondary 2 Chapter 5: Linear Graphs in Two Variables and Simultaneous Equations when coordinate plotting, axis reading or graph interpretation is unstable. Chapter 6 keeps the coordinate language but replaces constant-rate lines with changing-rate parabolas.

117. Specialist Companions

• Secondary 2 Mathematics Learning Guide | Quadratic Functions, Graphs and Turning Points
• Secondary 2 Mathematics Learning Guide | Quadratic Equations, Factorisation and Problem Solving
• Secondary 2 Mathematics Learning Guide | Algebraic Factorisation and Structural Control

118. Why This Chapter Matters for Chapter 7

Chapter 7 moves away from algebraic graphs and back into geometry: congruence, similarity and enlargement. But the same discipline continues. Equivalent forms become corresponding sides; scale factors replace algebraic multipliers; and conditions must still be checked before applying a rule. The graph-and-domain habit developed here will become scale-and-correspondence control in geometry.

119. Ready for Chapter 7?

You are ready to continue when the material appropriate to your current level is stable.

• Shared G2/G3: identify and manipulate quadratic expressions; factorise common, monic, selected non-unit and identity-pattern quadratics; verify by expansion.
• G3 Secondary 2 / G2 later-course: distinguish expression from equation; rearrange to zero; solve by factorisation and zero product; connect roots with x-intercepts; use the sign of the leading coefficient to predict opening; identify symmetry and turning points; sketch from structural features; interpret roots and turning points inside a model domain.

If one item is weak, return to the smallest section that owns it and solve a changed example. When the appropriate set is stable, continue to Chapter 7: Congruence, Similarity and Enlargement.