Question

1. sketch a graph of the function f(x)=x(x - 1)^3(x + 4)^2(x + 6). be sure to include a shaded x - axis.

Explanation:

Step1: Find the x - intercepts

Set $f(x)=0$. Then $x(x - 1)^3(x + 4)^2(x + 6)=0$. The solutions are $x = 0$, $x=1$ (with multiplicity 3), $x=-4$ (with multiplicity 2), and $x=-6$.

Step2: Determine the end - behavior

The leading term of the polynomial is obtained by multiplying the leading terms of each factor. The degree of the polynomial is $1 + 3+2 + 1=7$ (odd) and the leading coefficient is positive (since the product of the leading coefficients of each factor is positive). As $x\to-\infty$, $y\to-\infty$ and as $x\to+\infty$, $y\to+\infty$.

Step3: Analyze the behavior at the roots

• At $x = - 6$ (root of multiplicity 1), the graph crosses the x - axis.
• At $x=-4$ (root of multiplicity 2), the graph touches the x - axis and turns around.
• At $x = 0$ (root of multiplicity 1), the graph crosses the x - axis.
• At $x = 1$ (root of multiplicity 3), the graph crosses the x - axis and has a "flatter" behavior near $x = 1$ compared to a simple root.

Answer:

Sketch a graph with x - intercepts at $x=-6$, $x=-4$, $x = 0$, $x = 1$. The graph has the appropriate end - behavior (falls to the left and rises to the right) and the correct behavior at each root as described above. Shade the x - axis as required.