Question

graph the polynomial function $y = -x(x - 1)(x - 3)^2$.

Explanation:

Step1: Find x-intercepts

Set $y=0$, solve $-x(x-1)(x-3)^2=0$.
Solutions: $x=0$, $x=1$, $x=3$ (double root).

Step2: Determine end behavior

Degree of polynomial: $1+1+2=4$ (even), leading coefficient $-1$ (negative).
As $x\to+\infty$, $y\to-\infty$; as $x\to-\infty$, $y\to-\infty$.

Step3: Analyze intercept behavior

• At $x=0$: Crosses x-axis (odd multiplicity 1).
• At $x=1$: Crosses x-axis (odd multiplicity 1).
• At $x=3$: Touches x-axis (even multiplicity 2, turns around).

Step4: Calculate y-intercept

Set $x=0$: $y=-0(0-1)(0-3)^2=0$. Y-intercept is $(0,0)$.

Step5: Find test points

• For $x=2$: $y=-2(2-1)(2-3)^2=-2(1)(1)=-2$, point $(2,-2)$.
• For $x=4$: $y=-4(4-1)(4-3)^2=-4(3)(1)=-12$, point $(4,-12)$.
• For $x=-1$: $y=-(-1)(-1-1)(-1-3)^2=1(-2)(16)=-32$, point $(-1,-32)$.

Answer:

1. Plot intercepts: $(0,0)$, $(1,0)$, $(3,0)$
2. Sketch end behavior: Both ends of the graph point downward.
3. Draw the graph crossing the x-axis at $(0,0)$ and $(1,0)$, touching and turning around at $(3,0)$, passing through test points $(-1,-32)$, $(2,-2)$, $(4,-12)$, and connecting all points with a smooth, continuous curve.