1. **State the problem:** Analyze the given polynomial, quadratic, and linear functions within their specified domains, understand their behavior, and interpret key features based on the given points. 2. **Polynomial Functions:** The polynomial of degree 8 is defined as: $$f(x) = -0.002 \left(\frac{x}{3}\right)^8 + 0.05 \left(\frac{x}{3}\right)^6 - 0.4 \left(\frac{x}{3}\right)^4 + \left(\frac{x}{3}\right)^2$$ with domain $$-9.5 < x < 9.5$$. - Points: - At $$x = -9.50$$, $$y = 0.00$$ - At $$x = -4.75$$, $$y = 0.70$$ - At $$x = 0.00$$, $$y = 0.00$$ - At $$x = 4.75$$, $$y = 0.70$$ - At $$x = 9.50$$, $$y = 0.00$$ This polynomial has symmetrical behavior due to even powers, and values are zero or positive in these sample points, showing peaks around $$x=\pm4.75$$. 3. Another polynomial function is similar but shifted down by 1 unit: $$f(x) = -0.002 \left(\frac{x}{3}\right)^8 + 0.05 \left(\frac{x}{3}\right)^6 - 0.4 \left(\frac{x}{3}\right)^4 + \left(\frac{x}{3}\right)^2 - 1$$ with same domain and points: - $$y$$ values shifted down by 1 (e.g., $$y = -1.00, -0.30, -1.00, -0.30, -1.00$$) 4. There is a constant piecewise function: $$f(x) = 8$$ for $$-8.5 < x < -0.579$$ with constant $$y=8$$ at all points: - $$x = -8.50, -6.98, -5.46, -3.94, -0.58$$ - $$y = 8.00$$ for all listed $$x$$. 5. **Quadratic Functions:** Five parabolic functions are given in vertex form: $$f(x) = a(x + b)^2 + c$$ Each with given domain intervals and respective $$x, y$$ points showing maximum or minimum values consistent with the quadratic behavior. Examples: - $$f(x) = -0.8(x + 6)^2 + 10$$ for $$-7.38 < x < -4.748$$ - $$f(x) = -1.1(x + 4.1)^2 + 9.2$$ for $$-4.745 < x < -3.391$$ The negative leading coefficients indicate these are downward-opening parabolas, with peak $$y$$ values near the vertex. 6. **Linear Functions:** Four linear functions defined with specific domains and points: - $$f(x) = 0.6x + 13.1$$ for $$-8.5 < x < -8.18$$ - $$f(x) = -0.59x + 7.66$$ for $$-1 < x < -0.577$$ - $$f(x) = 3.7x - 13$$ for $$4.515 < x < 5.5$$ - $$f(x) = 0.3x + 5.7$$ for $$1 < x < 5.5$$ Each linear function shows constant rate of change within its domain, confirmed by the given points. **Summary:** The problem showcases understanding of polynomial degrees and symmetry, quadratic vertex forms and parabolic shapes, and linear rate of change through given data points and domains. Each function is carefully restricted to specific intervals, illustrating piecewise behavior and continuity where applicable.