GCSE Maths Algebra - How to Solve: Quadratic and Cubic Graphs (Sketching, Intercepts, Turning Points)

How to Solve: Quadratic and Cubic Graphs (Sketching, Intercepts, Turning Points)

Complete Guide For GCSE/A-Level Physics, Chemistry, Biology (Edexcel, AQA, OCR)

Introduction

"Mastering quadratic and cubic graphs doesn’t just get you 5–10 marks on your exam—it’s the key to predicting chemical reaction rates, modelling projectile motion in physics, and even understanding enzyme kinetics in biology. One sketch can save you 10 minutes of algebra in a time-pressured paper."

WHAT YOU NEED TO KNOW FIRST

Before starting, you must understand: 1. Factorising quadratics (e.g., x² – 5x + 6 = (x–2)(x–3)). 2. Solving linear equations (e.g., 2x + 3 = 0 → x = –1.5). 3. Basic coordinate geometry (plotting points, axes, intercepts).

If you’re shaky on any of these, pause and revise them first.

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Quadratic equation (roots)
   x = [–b ± √(b² – 4ac)] / 2a
2. a, b, c = coefficients in y = ax² + bx + c.

3. MEMORISE THIS (or use the quadratic formula on your exam sheet).

4. Turning point of a quadratic
   x = –b / 2a (x-coordinate of the vertex).

5. Substitute x back into y = ax² + bx + c to find y.

6. MEMORISE THIS.

7. y-intercept
   Always y = c (for y = ax² + bx + c) or y = d (for y = ax³ + bx² + cx + d).

8. Given on exam sheet (but you should know it anyway).

9. Factorised form (roots)
   y = a(x – p)(x – q) → roots at x = p and x = q.

10. MEMORISE THIS (helps sketching).

STEP-BY-STEP METHOD

For Quadratic Graphs (y = ax² + bx + c)

Step 1: Identify the shape - If a > 0 → U-shaped (opens upwards). - If a < 0 → ∩-shaped (opens downwards).

Step 2: Find the y-intercept - Set x = 0 → y = c. - Plot (0, c).

Step 3: Find the roots (x-intercepts) - Set y = 0 → solve ax² + bx + c = 0. - Use factorising, completing the square, or the quadratic formula. - Plot the roots (p, 0) and (q, 0).

Step 4: Find the turning point - Use x = –b / 2a to find the x-coordinate. - Substitute x back into the equation to find y. - Plot the turning point (x, y).

Step 5: Sketch the graph - Draw a smooth curve through the points: - y-intercept → roots → turning point. - Label all key points.

For Cubic Graphs (y = ax³ + bx² + cx + d)

Step 1: Identify the shape - If a > 0 → starts low, ends high (↗). - If a < 0 → starts high, ends low (↘).

Step 2: Find the y-intercept - Set x = 0 → y = d. - Plot (0, d).

Step 3: Find the roots (x-intercepts) - Set y = 0 → solve ax³ + bx² + cx + d = 0. - Try factorising (e.g., x = 1 or x = –2 as roots). - If stuck, use the factor theorem: f(k) = 0 → (x – k) is a factor. - Plot the roots (p, 0), (q, 0), (r, 0).

Step 4: Find turning points (optional for GCSE, required for A-Level) - Differentiate: dy/dx = 3ax² + 2bx + c. - Set dy/dx = 0 → solve for x. - Substitute x back into the original equation to find y. - Plot the turning points.

Step 5: Sketch the graph - Draw a smooth S-shaped curve through: - y-intercept → roots → turning points (if found). - Label all key points.

WORKED EXAMPLES

Example 1 – Basic Quadratic

Question: Sketch y = x² – 5x + 6.

Step 1: Shape - a = 1 (> 0) → U-shaped.

Step 2: y-intercept - x = 0 → y = 6. - Plot (0, 6).

Step 3: Roots - y = 0 → x² – 5x + 6 = 0. - Factorise: (x – 2)(x – 3) = 0 → x = 2 and x = 3. - Plot (2, 0) and (3, 0).

Step 4: Turning point - x = –b / 2a = –(–5) / 2(1) = 2.5. - y = (2.5)² – 5(2.5) + 6 = 6.25 – 12.5 + 6 = –0.25. - Plot (2.5, –0.25).

Step 5: Sketch - Draw a U-shape through (0, 6), (2, 0), (2.5, –0.25), (3, 0).

What we did and why: - Found the shape to know if it opens up or down. - Used the y-intercept for a starting point. - Solved for roots to know where it crosses the x-axis. - Calculated the turning point to find the vertex. - Connected the dots smoothly for an accurate sketch.

Example 2 – Medium Cubic

Question: Sketch y = x³ – 4x.

Step 1: Shape - a = 1 (> 0) → starts low, ends high.

Step 2: y-intercept - x = 0 → y = 0. - Plot (0, 0).

Step 3: Roots - y = 0 → x³ – 4x = 0. - Factorise: x(x² – 4) = 0 → x = 0 or x = ±2. - Plot (–2, 0), (0, 0), (2, 0).

Step 4: Turning points (A-Level only) - dy/dx = 3x² – 4. - Set dy/dx = 0 → 3x² – 4 = 0 → x = ±√(4/3) ≈ ±1.15. - y at x = 1.15: (1.15)³ – 4(1.15) ≈ –3.08. - y at x = –1.15: (–1.15)³ – 4(–1.15) ≈ 3.08. - Plot (1.15, –3.08) and (–1.15, 3.08).

Step 5: Sketch - Draw an S-shape through (–2, 0), (–1.15, 3.08), (0, 0), (1.15, –3.08), (2, 0).

What we did and why: - Identified the cubic’s direction from the a coefficient. - Found the y-intercept for a reference point. - Solved for roots to know where it crosses the x-axis. - Used calculus (A-Level) to find turning points for accuracy. - Connected the points smoothly for the correct S-shape.

Example 3 – Exam-Style (Disguised)

Question: A quadratic graph has roots at x = –1 and x = 3 and passes through (0, –6). Sketch the graph.

Step 1: Write the equation in factorised form - Roots at x = –1 and x = 3 → y = a(x + 1)(x – 3).

Step 2: Find a using the given point - Passes through (0, –6) → –6 = a(0 + 1)(0 – 3) → –6 = a(1)(–3) → a = 2. - Equation: y = 2(x + 1)(x – 3).

Step 3: Expand to standard form (optional) - y = 2(x² – 2x – 3) → y = 2x² – 4x – 6.

Step 4: Shape - a = 2 (> 0) → U-shaped.

Step 5: y-intercept - x = 0 → y = –6. - Plot (0, –6).

Step 6: Roots - Already given: (–1, 0) and (3, 0).

Step 7: Turning point - x = –b / 2a = –(–4) / 2(2) = 1. - y = 2(1)² – 4(1) – 6 = 2 – 4 – 6 = –8. - Plot (1, –8).

Step 8: Sketch - Draw a U-shape through (–1, 0), (0, –6), (1, –8), (3, 0).

What we did and why: - Used roots to write the factorised form. - Found a using a given point to complete the equation. - Expanded (optional) to confirm standard form. - Followed the same steps as before to sketch accurately.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Here’s the night-before cheat sheet for quadratic and cubic graphs: 1. Quadratics (U/∩ shape):
- a > 0 → U, a < 0 → ∩.
- y-intercept: x = 0 → y = c.
- Roots: solve y = 0 (factorise or use quadratic formula).
- Turning point: x = –b / 2a, then find y. 2. Cubics (S shape):
- a > 0 → starts low, ends high; a < 0 → opposite.
- y-intercept: x = 0 → y = d.
- Roots: factorise or use factor theorem.
- Turning points (A-Level): differentiate and set dy/dx = 0. 3. Sketching:
- Plot y-intercept, roots, turning points.
- Draw smooth curves—no straight lines! 4. Exam traps:
- Disguised roots? Write y = a(x – p)(x – q) first.
- Repeated roots? Graph touches but doesn’t cross.
- Non-integer turning points? Use the formula and substitute carefully.

Now go sketch like a pro—you’ve got this!"