1. **Problem Statement:** Sketch and analyze the curves for the following functions: a) $y = x^2 (x + 2)$ c) $y = (x + 1)^2 (x - 2)$ a) $y = |(x + 1)(x - 2)(x - 3)|$ b) $y = x^2(5 - 2x)$ d) $y = (x - 2)^2(10 - 3x)$ 2. **General Approach:** - Identify zeros (roots) by setting the function equal to zero. - Determine multiplicity of roots to understand if the graph touches or crosses the x-axis. - Analyze end behavior by considering the degree and leading coefficient. - For absolute value functions, reflect negative parts above the x-axis. 3. **Function a) $y = x^2 (x + 2)$:** - Roots: $x=0$ (multiplicity 2), $x=-2$ (multiplicity 1). - At $x=0$, the graph touches the x-axis and turns around (because of even multiplicity). - At $x=-2$, the graph crosses the x-axis. - Degree: 3 (odd), leading term $x^3$, so as $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$. 4. **Function c) $y = (x + 1)^2 (x - 2)$:** - Roots: $x=-1$ (multiplicity 2), $x=2$ (multiplicity 1). - At $x=-1$, graph touches and turns. - At $x=2$, graph crosses. - Degree: 3 (odd), leading term $x^3$, same end behavior as above. 5. **Function a) $y = |(x + 1)(x - 2)(x - 3)|$:** - Roots: $x=-1, 2, 3$. - Without absolute value, graph crosses x-axis at these points. - Absolute value reflects any negative values above x-axis, so graph never goes below x-axis. - Degree: 3 (odd), but graph is always $\geq 0$. 6. **Function b) $y = x^2(5 - 2x)$:** - Roots: $x=0$ (multiplicity 2), $x=\frac{5}{2}$ (multiplicity 1). - At $x=0$, graph touches and turns. - At $x=\frac{5}{2}$, graph crosses. - Degree: 3 (odd), leading term $-2x^3$, so as $x \to \infty$, $y \to -\infty$; as $x \to -\infty$, $y \to \infty$. 7. **Function d) $y = (x - 2)^2(10 - 3x)$:** - Roots: $x=2$ (multiplicity 2), $x=\frac{10}{3}$ (multiplicity 1). - At $x=2$, graph touches and turns. - At $x=\frac{10}{3}$, graph crosses. - Degree: 3 (odd), leading term $-3x^3$, same end behavior as function b). **Summary:** - For each polynomial, roots and their multiplicities determine x-intercept behavior. - Odd degree polynomials have opposite end behaviors depending on leading coefficient sign. - Absolute value modifies the graph by reflecting negative parts above x-axis. Final answers are the sketches and analysis as described above.