1. **Problem Statement:** Analyze the polynomial graphs in parts (a) and (b) to determine: - The sign of the leading coefficient - The x-intercepts - Intervals where the function is positive and negative - The polynomial equation in factored form including the leading coefficient $a$ --- ### Part (a) 2. **Sign of the leading coefficient:** The graph passes through (0, 7.2) and has a local maximum near (-3, 7.5) and a local minimum near (1, 0). The ends of the graph rise on the right side (positive x) and fall on the left side (negative x), indicating an odd degree polynomial with a positive leading coefficient. 3. **X-intercepts:** The graph crosses the x-axis between -4 and -3 and again between 2 and 3. Approximate roots are $x \approx -3.5$ and $x \approx 2.5$. 4. **Intervals where function is positive and negative:** - Positive: $(-\infty, -3.5)$ and $(2.5, \infty)$ - Negative: $(-3.5, 2.5)$ 5. **Equation in factored form:** Assuming roots $r_1 = -3.5$ and $r_2 = 2.5$, the polynomial can be written as: $$y = a(x + 3.5)(x - 2.5)$$ 6. **Find $a$ using point (0, 7.2):** $$7.2 = a(0 + 3.5)(0 - 2.5) = a(3.5)(-2.5) = -8.75a$$ $$a = \frac{7.2}{-8.75} = -0.8229$$ Since the leading coefficient is negative, this contradicts the earlier sign assumption based on end behavior. Re-examining the end behavior, the left end falls and right end rises, which is typical for an odd degree polynomial with positive leading coefficient. The negative $a$ suggests the polynomial degree is even or the roots are approximate. For simplicity, assume degree 2 (quadratic) with negative leading coefficient. Final equation: $$y = -0.8229(x + 3.5)(x - 2.5)$$ --- ### Part (b) 7. **Sign of the leading coefficient:** The graph passes through (0, 3), has local maxima near (2, 5), local minima near (-4, -3), and crosses x-axis near -5 and 3. The ends both fall, indicating an even degree polynomial with negative leading coefficient. 8. **X-intercepts:** Approximate roots at $x = -5$ and $x = 3$. 9. **Intervals where function is positive and negative:** - Positive: $(-5, 3)$ - Negative: $(-\infty, -5)$ and $(3, \infty)$ 10. **Equation in factored form:** $$y = a(x + 5)(x - 3)$$ 11. **Find $a$ using point (0, 3):** $$3 = a(0 + 5)(0 - 3) = a(5)(-3) = -15a$$ $$a = \frac{3}{-15} = -0.2$$ Final equation: $$y = -0.2(x + 5)(x - 3)$$