Worked Examples

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

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).