1. **Problem Statement:** We need to sketch the graphs of two functions: - $f(x) = (x - 1)^2 (x + 1)^3$ - $f(x) = 2x^2 - 4x - 2$ 2. **Function 1: $f(x) = (x - 1)^2 (x + 1)^3$** - This is a polynomial function formed by the product of two factors. - The zeros (roots) are at $x=1$ and $x=-1$. - The multiplicity of the root at $x=1$ is 2 (even), so the graph touches the x-axis and turns around there. - The multiplicity of the root at $x=-1$ is 3 (odd), so the graph crosses the x-axis and flattens out somewhat at this root. 3. **Behavior at roots:** - At $x=1$, since multiplicity is even, the graph touches the x-axis and bounces back. - At $x=-1$, since multiplicity is odd, the graph crosses the x-axis. 4. **End behavior:** - The degree of the polynomial is $2 + 3 = 5$ (odd degree). - The leading term is $x^2 imes x^3 = x^5$, so as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$. 5. **Function 2: $f(x) = 2x^2 - 4x - 2$** - This is a quadratic function. - The standard form is $ax^2 + bx + c$ with $a=2$, $b=-4$, $c=-2$. 6. **Vertex and axis of symmetry:** - Axis of symmetry: $x = -\frac{b}{2a} = -\frac{-4}{2 \times 2} = 1$ - Vertex: $f(1) = 2(1)^2 - 4(1) - 2 = 2 - 4 - 2 = -4$ 7. **Roots:** - Use quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ - Calculate discriminant: $\Delta = (-4)^2 - 4 \times 2 \times (-2) = 16 + 16 = 32$ - Roots: $x = \frac{4 \pm \sqrt{32}}{4} = \frac{4 \pm 4\sqrt{2}}{4} = 1 \pm \sqrt{2}$ 8. **Graph features:** - Parabola opens upward since $a=2 > 0$. - Vertex at $(1, -4)$ is the minimum point. - Roots at $x = 1 - \sqrt{2}$ and $x = 1 + \sqrt{2}$. **Final answers:** - The graph of $f(x) = (x - 1)^2 (x + 1)^3$ has roots at $x=-1$ (crossing) and $x=1$ (touching), with end behavior going to $-\infty$ as $x \to -\infty$ and $\infty$ as $x \to \infty$. - The graph of $f(x) = 2x^2 - 4x - 2$ is a parabola opening upward with vertex at $(1, -4)$ and roots at $1 \pm \sqrt{2}$.