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 Maths

Introduction

"Mastering quadratic and cubic graphs unlocks 10–15% of your GCSE/A-Level Maths exam marks—think intercepts, turning points, and sketching curves in seconds. This is how you turn ‘I don’t get it’ into ‘I’ve got this’ on exam day."

What You Need To Know First

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, y-intercept).

Key Vocabulary

Formulas To Know

1. Quadratic in standard form
   y = ax² + bx + c
2. a, b, c = constants (a ≠ 0).

3. MEMORISE THIS: The y-intercept is always (0, c).

4. Quadratic in factorised form
   y = a(x – p)(x – q)

5. p, q = roots (x-intercepts).

6. MEMORISE THIS: The turning point’s x-coordinate is halfway between p and q.

7. Discriminant (for roots)
   Δ = b² – 4ac

8. MEMORISE THIS:

   • Δ > 0 → 2 real roots.
   • Δ = 0 → 1 real root (repeated).
   • Δ < 0 → No real roots.

9. Turning point of a quadratic
   x = –b / (2a)

10. MEMORISE THIS: Plug this x back into the equation to find the y-coordinate.

11. Cubic in standard form
    y = ax³ + bx² + cx + d

12. a, b, c, d = constants (a ≠ 0).
13. MEMORISE THIS: The y-intercept is (0, d).

Step-by-Step Method

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

1. Find the y-intercept: Set x = 0. Write as (0, c).
2. Find the x-intercepts (roots): Solve y = 0 (factorise or use the quadratic formula).
3. If Δ < 0, there are no x-intercepts (graph doesn’t cross the x-axis).
4. Find the turning point:
5. x-coordinate: x = –b / (2a).
6. y-coordinate: Plug x back into the equation.
7. Determine the shape:
8. a > 0 → U-shaped (minimum turning point).
9. a < 0 → ∩-shaped (maximum turning point).
10. Sketch the graph:
11. Plot the y-intercept.
12. Plot the x-intercepts (if they exist).
13. Plot the turning point.
14. Draw a smooth curve through all points.

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

1. Find the y-intercept: Set x = 0. Write as (0, d).
2. Find the x-intercepts (roots): Solve y = 0 (factorise if possible).
3. If factorising is hard, use the factor theorem (try x = ±1, ±2, etc.).
4. Determine the end behaviour:
5. a > 0 → Graph falls to the left, rises to the right.
6. a < 0 → Graph rises to the left, falls to the right.
7. Find turning points (optional for GCSE, required for A-Level):
8. Differentiate to find dy/dx = 3ax² + 2bx + c.
9. Set dy/dx = 0 and solve for x.
10. Plug x back into the original equation to find y.
11. Sketch the graph:
12. Plot the y-intercept.
13. Plot the x-intercepts.
14. Plot turning points (if found).
15. Draw a smooth curve with the correct end behaviour.

Worked Examples

Example 1 – Basic Quadratic

Sketch y = x² – 5x + 6. Label intercepts and turning point.

Step 1: y-intercept Set x = 0: y = 0 – 0 + 6 = 6 → (0, 6)

Step 2: x-intercepts Solve x² – 5x + 6 = 0 Factorise: (x – 2)(x – 3) = 0 → x = 2 or x = 3 → (2, 0) and (3, 0)

Step 3: Turning point a = 1, b = –5 x = –b / (2a) = –(–5) / 2 = 2.5 Plug x = 2.5 into y: y = (2.5)² – 5(2.5) + 6 = 6.25 – 12.5 + 6 = –0.25 → (2.5, –0.25)

Step 4: Shape a = 1 > 0 → U-shaped (minimum turning point).

Step 5: Sketch - Plot (0, 6), (2, 0), (3, 0), (2.5, –0.25). - Draw a smooth U-shaped curve through all points.

What we did and why: We found all key points to plot the graph accurately. The turning point tells us the lowest point on the curve.

Example 2 – Medium Quadratic (No Real Roots)

Sketch y = –x² + 4x – 5. Label intercepts and turning point.

Step 1: y-intercept Set x = 0: y = –0 + 0 – 5 = –5 → (0, –5)

Step 2: x-intercepts Solve –x² + 4x – 5 = 0 Multiply by –1: x² – 4x + 5 = 0 Δ = b² – 4ac = 16 – 20 = –4 < 0 → No real roots (graph doesn’t cross x-axis).

Step 3: Turning point a = –1, b = 4 x = –b / (2a) = –4 / (–2) = 2 Plug x = 2 into y: y = –(2)² + 4(2) – 5 = –4 + 8 – 5 = –1 → (2, –1)

Step 4: Shape a = –1 < 0 → ∩-shaped (maximum turning point).

Step 5: Sketch - Plot (0, –5) and (2, –1). - Draw a ∩-shaped curve above the x-axis (since no x-intercepts).

What we did and why: Even with no x-intercepts, we used the y-intercept and turning point to sketch the graph. The discriminant told us the graph doesn’t cross the x-axis.

Example 3 – Exam-Style Cubic

Sketch y = (x + 1)(x – 2)(x – 3). Label intercepts and state the end behaviour.

Step 1: y-intercept Set x = 0: y = (1)(–2)(–3) = 6 → (0, 6)

Step 2: x-intercepts Solve (x + 1)(x – 2)(x – 3) = 0 → x = –1, x = 2, x = 3 → (–1, 0), (2, 0), (3, 0)

Step 3: End behaviour Expand the first term: y = x³ + ... (a = 1 > 0) → Falls to the left, rises to the right.

Step 4: Turning points (A-Level only) Differentiate: dy/dx = 3x² – 8x + 1 Set dy/dx = 0: 3x² – 8x + 1 = 0 Use quadratic formula: x = [8 ± √(64 – 12)] / 6 = [8 ± √52] / 6 ≈ 0.13 or 2.54 Plug back into y to find y-coordinates (optional for sketching).

Step 5: Sketch - Plot (0, 6), (–1, 0), (2, 0), (3, 0). - Draw a smooth curve starting from the bottom left, passing through x-intercepts, and rising to the top right.

What we did and why: We used the factorised form to quickly find intercepts. The end behaviour tells us the general shape without needing to plot every point.

Common Mistakes

1. Mistake: Forgetting the y-intercept.
   WHY IT HAPPENS: Students focus on x-intercepts and turning points.
   CORRECT APPROACH: Always set x = 0 first.

2. Mistake: Mixing up the shape of quadratics (U vs. ∩).
   WHY IT HAPPENS: Not checking the sign of a.
   CORRECT APPROACH: a > 0 → U-shaped; a < 0 → ∩-shaped.

3. Mistake: Incorrectly calculating the turning point.
   WHY IT HAPPENS: Using the wrong formula or miscalculating –b / (2a).
   CORRECT APPROACH: Double-check the formula and arithmetic.

4. Mistake: Assuming all cubics have 3 x-intercepts.
   WHY IT HAPPENS: Not considering repeated roots or no real roots.
   CORRECT APPROACH: Solve y = 0 to confirm.

5. Mistake: Drawing straight lines between points.
   WHY IT HAPPENS: Not understanding the smooth nature of polynomials.
   CORRECT APPROACH: Always draw smooth curves, not sharp corners.

Exam Traps

1. Trap: The quadratic has no x-intercepts, but the question asks for them.
   How to Spot it: The discriminant is negative (Δ < 0).
   How to Avoid it: Write "No real roots" and move on.

2. Trap: The cubic is given in expanded form, but factorising is easier.
   How to Spot it: The question says "sketch" but gives y = x³ + ....
   How to Avoid it: Try the factor theorem (x = ±1, ±2, etc.) before expanding.

3. Trap: The turning point is not a whole number, but the question expects an exact answer.
   How to Spot it: The x-coordinate is a fraction or surd.
   How to Avoid it: Leave the answer as a fraction or simplified surd (e.g., x = 3/2 or x = √5).

1-Minute Recap

"Here’s the night-before cheat sheet for quadratic and cubic graphs: 1. Quadratics:
- y-intercept: (0, c).
- x-intercepts: Solve y = 0 (factorise or use the quadratic formula).
- Turning point: x = –b / (2a), then plug back to find y.
- Shape: a > 0 → U; a < 0 → ∩. 2. Cubics:
- y-intercept: (0, d).
- x-intercepts: Solve y = 0 (factorise if possible).
- End behaviour: a > 0 → falls left, rises right; a < 0 → rises left, falls right. 3. Always:
- Plot intercepts first.
- Draw smooth curves, not straight lines.
- Label everything clearly. You’ve got this—go smash that exam!