This study notes covers Worked Examples within Cubic Graphs for GCSE Mathematics. Revise Cubic Graphs in Graphs for GCSE Mathematics with 11 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 8 of 10 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 8 of 10
Practice
11 questions
Recall
10 flashcards
✏️ Worked Examples
Example 1: Sketching a Cubic from Factorised Form
Question: Sketch y = (x + 1)(x - 2)(x - 3), clearly showing all intercepts.
Show Solution
Step 1: Find the roots — set each bracket to zero:
x + 1 = 0 → x = -1 x - 2 = 0 → x = 2 x - 3 = 0 → x = 3
Roots at x = -1, 2, 3
Step 2: Find the y-intercept — set x = 0:
y = (0 + 1)(0 - 2)(0 - 3) = 1 × (-2) × (-3) = 6 → Point (0, 6)
Step 3: Identify the shape — leading term is x³ (positive), so positive cubic: rises from bottom-left to top-right.
Step 4: Sketch — the curve enters bottom-left, crosses x-axis at x = -1 (rises above), crosses at x = 2 (dips below), crosses at x = 3, and rises to top-right. Passes through (0, 6).
Answer: Roots at x = -1, 2, 3; y-intercept at (0, 6); positive cubic shape
Example 2: Table of Values for a Cubic Graph
Question: Complete the table of values for y = x³ - 3x + 1 for x from -2 to 2.
Show Solution
Calculate y for each x value:
x = -2: y = (-8) - 3(-2) + 1 = -8 + 6 + 1 = -1
x = -1: y = (-1) - 3(-1) + 1 = -1 + 3 + 1 = 3
x = 0: y = 0 - 0 + 1 = 1
x = 1: y = 1 - 3 + 1 = -1
x = 2: y = 8 - 6 + 1 = 3
Features: local maximum near x = -1 (y = 3), local minimum near x = 1 (y = -1), positive cubic shape.
Answer: y values: -1, 3, 1, -1, 3. Positive cubic with local max ≈ (-1, 3) and local min ≈ (1, -1).