Example 1: Sketching a Cubic from Factorised Form
Question: Sketch y = (x + 1)(x - 2)(x - 3), clearly showing all intercepts.
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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.
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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).
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.
Practice questions for Cubic Graphs
Which of the following best describes the general shape of the graph y = x³?
Explain how you can tell from the equation of a cubic whether its graph rises or falls as x approaches positive infinity.