1. **Stating the problem:** We are given points $(-2,-1)$, $(-1,0)$, $(0,-1)$, $(1,-1)$, and $(2,3)$ and need to analyze or find a function that fits these points. 2. **Approach:** Since we have 5 points, one way is to find a polynomial of degree 4 that passes through all these points. 3. **General form:** A polynomial of degree 4 is given by $$y = ax^4 + bx^3 + cx^2 + dx + e$$ 4. **Set up equations:** Substitute each point into the polynomial: - For $(-2,-1)$: $$-1 = a(-2)^4 + b(-2)^3 + c(-2)^2 + d(-2) + e = 16a - 8b + 4c - 2d + e$$ - For $(-1,0)$: $$0 = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e = a - b + c - d + e$$ - For $(0,-1)$: $$-1 = e$$ - For $(1,-1)$: $$-1 = a + b + c + d + e$$ - For $(2,3)$: $$3 = 16a + 8b + 4c + 2d + e$$ 5. **From point $(0,-1)$, we get:** $$e = -1$$ 6. **Substitute $e = -1$ into other equations:** - $-1 = 16a - 8b + 4c - 2d - 1 \\ \Rightarrow 0 = 16a - 8b + 4c - 2d$ - $0 = a - b + c - d - 1 \\ \Rightarrow 1 = a - b + c - d$ - $-1 = a + b + c + d - 1 \\ \Rightarrow 0 = a + b + c + d$ - $3 = 16a + 8b + 4c + 2d - 1 \\ \Rightarrow 4 = 16a + 8b + 4c + 2d$ 7. **Rewrite system:** $$\begin{cases} 16a - 8b + 4c - 2d = 0 \\ a - b + c - d = 1 \\ a + b + c + d = 0 \\ 16a + 8b + 4c + 2d = 4 \end{cases}$$ 8. **Add first and fourth equations:** $$ (16a - 8b + 4c - 2d) + (16a + 8b + 4c + 2d) = 0 + 4 \\ 32a + 8c = 4 \\ 4a + c = \frac{1}{8}$$ 9. **Add second and third equations:** $$ (a - b + c - d) + (a + b + c + d) = 1 + 0 \\ 2a + 2c = 1 \\ a + c = \frac{1}{2}$$ 10. **From step 8 and 9:** $$4a + c = \frac{1}{8} \\ a + c = \frac{1}{2}$$ Subtract second from first: $$ (4a + c) - (a + c) = \frac{1}{8} - \frac{1}{2} \\ 3a = -\frac{3}{8} \\ a = -\frac{1}{8}$$ 11. **Find $c$:** $$ a + c = \frac{1}{2} \\ -\frac{1}{8} + c = \frac{1}{2} \\ c = \frac{1}{2} + \frac{1}{8} = \frac{5}{8}$$ 12. **Use $a$ and $c$ in second equation:** $$ a - b + c - d = 1 \\ -\frac{1}{8} - b + \frac{5}{8} - d = 1 \\ \frac{4}{8} - b - d = 1 \\ -b - d = 1 - \frac{1}{2} = \frac{1}{2} \\ b + d = -\frac{1}{2}$$ 13. **Use $a$ and $c$ in third equation:** $$ a + b + c + d = 0 \\ -\frac{1}{8} + b + \frac{5}{8} + d = 0 \\ \frac{4}{8} + b + d = 0 \\ b + d = -\frac{1}{2}$$ 14. **From steps 12 and 13, $b + d = -\frac{1}{2}$ consistent.** 15. **Use first equation:** $$16a - 8b + 4c - 2d = 0 \\ 16(-\frac{1}{8}) - 8b + 4(\frac{5}{8}) - 2d = 0 \\ -2 - 8b + 2.5 - 2d = 0 \\ 0.5 - 8b - 2d = 0 \\ -8b - 2d = -0.5$$ 16. **Divide entire equation by -2:** $$ \cancel{-8b} \cancel{-2d} = \cancel{-0.5} \\ 4b + d = 0.25$$ 17. **From $b + d = -0.5$, express $d = -0.5 - b$ and substitute into $4b + d = 0.25$:** $$4b + (-0.5 - b) = 0.25 \\ 3b - 0.5 = 0.25 \\ 3b = 0.75 \\ b = 0.25$$ 18. **Find $d$:** $$ d = -0.5 - b = -0.5 - 0.25 = -0.75$$ 19. **Summary of coefficients:** $$a = -\frac{1}{8}, b = \frac{1}{4}, c = \frac{5}{8}, d = -\frac{3}{4}, e = -1$$ 20. **Final polynomial:** $$y = -\frac{1}{8}x^4 + \frac{1}{4}x^3 + \frac{5}{8}x^2 - \frac{3}{4}x - 1$$ This polynomial passes through all given points.