1. Finding where a cubic function crosses the y-axis
Solving the equation of the y-axis (x = 0) with the equation of the cubic function y = ax 3 + bx 2 + cx + d gives y = d. The graph of the cubic function crosses the y-axis at (0, d ).
EXAMPLE 3
Find where the following cubic functions cross the y-axis:
(a) y = −2x 3 + 3x 2 − 5x − 7 (b) y = x 3 + 11x 2 − 3x + 4
Solution (a) y = −2x 3 + 3x 2 − 5x − 7 x = 0: y = −7 It crosses the y-axis at (0, −7).
(b) y = x 3 + 11x 2 − 3x + 4 x = 0: y = 4 It crosses the y-axis at (0, 4).
2. Finding where a cubic function crosses
the x-axis Solving the equation of the x-axis (y = 0) with the cubic function y = ax 3 + bx 2 + cx + d gives ax 3 + bx 2 + cx + d = 0.
The point(s) where the graph crosses the x-axis are the solutions of the cubic equation ax 3 + bx 2 + cx + d = 0.
EXAMPLE 4
Draw a sketch of y = (2x + 1)(x − 3)(x − 1), x ∈ R, by fi nding where it crosses the axes.
Solution y = (2x + 1)(x − 3)(x − 1) Crosses x-axis: y = 0: (2x + 1)(x − 3)(x − 1) = 0 x = − 1
_ 2 , 1, 3
Crosses y-axis: x = 0: y = (+1)(−3)(−1) = +3 It crosses the y-axis at (0, 3).
If you multiply out the brackets, you get +2x 3 as the fi rst term in the cubic function.
a = +2 > 0 [This gives us the general shape.] y
