1. **Problem statement:** We want to find a cubic function $f(x) = ax^3 + bx^2 + cx + d$ that passes through the points $P_1(0, -1)$, $P_2(1, 1)$, $P_3(-1, -7)$, and $P_4(\frac{1}{2}, -\frac{5}{8})$. 2. **Set up equations using the points:** Substitute each point into the function $f(x)$: - For $P_1(0, -1)$: $$a\cdot0^3 + b\cdot0^2 + c\cdot0 + d = -1 \implies d = -1$$ - For $P_2(1, 1)$: $$a\cdot1^3 + b\cdot1^2 + c\cdot1 + d = 1 \implies a + b + c + d = 1$$ - For $P_3(-1, -7)$: $$a(-1)^3 + b(-1)^2 + c(-1) + d = -7 \implies -a + b - c + d = -7$$ - For $P_4(\frac{1}{2}, -\frac{5}{8})$: $$a\left(\frac{1}{2}\right)^3 + b\left(\frac{1}{2}\right)^2 + c\left(\frac{1}{2}\right) + d = -\frac{5}{8}$$ Simplify the last equation: $$a\cdot\frac{1}{8} + b\cdot\frac{1}{4} + c\cdot\frac{1}{2} + d = -\frac{5}{8}$$ 3. **Substitute $d = -1$ into the other equations:** - From $P_2$: $$a + b + c - 1 = 1 \implies a + b + c = 2$$ - From $P_3$: $$-a + b - c - 1 = -7 \implies -a + b - c = -6$$ - From $P_4$: $$\frac{a}{8} + \frac{b}{4} + \frac{c}{2} - 1 = -\frac{5}{8} \implies \frac{a}{8} + \frac{b}{4} + \frac{c}{2} = \frac{3}{8}$$ 4. **Solve the system of three equations:** $$\begin{cases} a + b + c = 2 \\ -a + b - c = -6 \\ \frac{a}{8} + \frac{b}{4} + \frac{c}{2} = \frac{3}{8} \end{cases}$$ Add the first two equations to eliminate $a$ and $c$: $$ (a + b + c) + (-a + b - c) = 2 + (-6) \implies 2b = -4 \implies b = -2$$ Substitute $b = -2$ into the first equation: $$a - 2 + c = 2 \implies a + c = 4$$ Substitute $b = -2$ into the third equation: $$\frac{a}{8} + \frac{-2}{4} + \frac{c}{2} = \frac{3}{8} \implies \frac{a}{8} - \frac{1}{2} + \frac{c}{2} = \frac{3}{8}$$ Add $\frac{1}{2}$ to both sides: $$\frac{a}{8} + \frac{c}{2} = \frac{3}{8} + \frac{4}{8} = \frac{7}{8}$$ Multiply both sides by 8: $$a + 4c = 7$$ 5. **Solve the system:** $$\begin{cases} a + c = 4 \\ a + 4c = 7 \end{cases}$$ Subtract the first from the second: $$ (a + 4c) - (a + c) = 7 - 4 \implies 3c = 3 \implies c = 1$$ Substitute $c = 1$ into $a + c = 4$: $$a + 1 = 4 \implies a = 3$$ 6. **Final function:** $$f(x) = 3x^3 - 2x^2 + x - 1$$ This cubic function passes through all four given points.