1. **State the problem:** We need to find the equation of a cubic function $y = ax^3 + bx^2 + cx + d$ that passes through the points $(2.5, 0)$, $(3.5, 2)$, and $(-3, -1)$. Since a cubic has 4 coefficients, we need 4 conditions. We assume the cubic passes through the origin $(0,0)$ as the fourth point since it is not given explicitly but often assumed for such problems. 2. **Set up the system of equations:** Using the general cubic form $y = ax^3 + bx^2 + cx + d$, plug in each point: - At $(0,0)$: $0 = a\cdot0^3 + b\cdot0^2 + c\cdot0 + d \Rightarrow d = 0$ - At $(2.5,0)$: $0 = a(2.5)^3 + b(2.5)^2 + c(2.5) + d$ - At $(3.5,2)$: $2 = a(3.5)^3 + b(3.5)^2 + c(3.5) + d$ - At $(-3,-1)$: $-1 = a(-3)^3 + b(-3)^2 + c(-3) + d$ Since $d=0$, simplify: 3. **Write the equations explicitly:** - $0 = a(15.625) + b(6.25) + c(2.5)$ - $2 = a(42.875) + b(12.25) + c(3.5)$ - $-1 = a(-27) + b(9) + c(-3)$ 4. **Solve the system:** From the first equation: $$0 = 15.625a + 6.25b + 2.5c$$ Divide by 2.5: $$0 = \cancel{2.5}(6.25a) + \cancel{2.5}(2.5b) + \cancel{2.5}c \Rightarrow 0 = 6.25a + 2.5b + c$$ Express $c$: $$c = -6.25a - 2.5b$$ Substitute $c$ into the other two equations: Second equation: $$2 = 42.875a + 12.25b + 3.5c = 42.875a + 12.25b + 3.5(-6.25a - 2.5b)$$ $$2 = 42.875a + 12.25b - 21.875a - 8.75b = (42.875 - 21.875)a + (12.25 - 8.75)b = 21a + 3.5b$$ Third equation: $$-1 = -27a + 9b - 3c = -27a + 9b - 3(-6.25a - 2.5b)$$ $$-1 = -27a + 9b + 18.75a + 7.5b = (-27 + 18.75)a + (9 + 7.5)b = -8.25a + 16.5b$$ 5. **Solve the two equations:** $$2 = 21a + 3.5b$$ $$-1 = -8.25a + 16.5b$$ Multiply the first equation by 16.5 and the second by 3.5 to eliminate $b$: $$2 \times 16.5 = 33a + 57.75b$$ $$-1 \times 3.5 = -28.875a + 57.75b$$ Subtract second from first: $$33a + 57.75b - (-28.875a + 57.75b) = 33a + 57.75b + 28.875a - 57.75b = 61.875a = 33 - (-3.5) = 35.5$$ So: $$61.875a = 35.5 \Rightarrow a = \frac{35.5}{61.875} = \frac{142}{247.5} = \frac{284}{495} \approx 0.5737$$ Substitute $a$ back into $2 = 21a + 3.5b$: $$2 = 21(0.5737) + 3.5b \Rightarrow 2 = 12.0477 + 3.5b \Rightarrow 3.5b = 2 - 12.0477 = -10.0477$$ $$b = \frac{-10.0477}{3.5} \approx -2.871$$(rounded) Calculate $c$: $$c = -6.25a - 2.5b = -6.25(0.5737) - 2.5(-2.871) = -3.5856 + 7.1775 = 3.5919$$ 6. **Write the final cubic function:** $$y = 0.5737x^3 - 2.871x^2 + 3.5919x$$ This is the cubic function passing through the given points and the origin. **Final answer:** $$y \approx 0.574x^3 - 2.871x^2 + 3.592x$$