summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y =…

Explanation:

Step1: Find the x - intercepts

Set $f(x)=0$, so $2x(x - 3)^3=0$. Then $x = 0$ or $x=3$.

Step2: Find the y - intercept

Set $x = 0$ in $f(x)$, then $y=f(0)=2\times0\times(0 - 3)^3=0$.

Step3: Analyze the end - behavior

The degree of the polynomial $y = 2x(x - 3)^3=2x(x^{3}-9x^{2}+27x - 27)=2x^{4}-18x^{3}+54x^{2}-54x$ is $n = 4$ (even) and the leading coefficient $a = 2>0$. So $\lim_{x\rightarrow\pm\infty}f(x)=+\infty$.

Step4: Analyze the multiplicity of the roots

The root $x = 0$ has multiplicity $m_1=1$ (crosses the x - axis), and the root $x = 3$ has multiplicity $m_2=3$ (crosses the x - axis).

Answer:

Based on the above - mentioned analysis of x - intercepts, y - intercept, end - behavior and multiplicity of roots, we can determine the correct graph. Without seeing the actual shapes of the graphs in detail from the reference, but using the above properties: The function crosses the x - axis at $x = 0$ and $x = 3$ and has an even - degree with positive leading coefficient. You need to check which of the given graphs A, B, C, D satisfy these properties.