it goes through the orgin and ponts are (-2,-3) (1,.75) (2,3)
thanks
Passes through (0, 0), symmetric and no other x-intercepts: $\displaystyle y = ax(x^2 + b) \Rightarrow y = ax^3 + abx \Rightarrow y = ax^3 + cx$.
Sub (2, 3): $\displaystyle 3 = a(2)^3 + 2c \Rightarrow 3 = 8a + 2c$ .... (1)
Sub (1, 3/4): $\displaystyle \frac{3}{4} = a + c$ .... (2)
Solve (1) and (2) simultaneously (use elimination method):
(1) - 2 (2): $\displaystyle 3 - \frac{3}{2} = 8a + 2c - 2a - 2c \Rightarrow \frac{3}{2} = 6a \Rightarrow a = \frac{3}{12} = \frac{1}{4}$.
Therefore $\displaystyle c = \frac{1}{2}$.
Therefore: $\displaystyle y = \frac{x^3}{4} + \frac{x}{2} = \frac{1}{4} (x^3 + 2x)$