Let's analyze the two functions provided in the problem:

1. $g(x) = 2x^3 - 4x^2 - 6x$
2. $h(x) = -(x - 4)^3(x - 1)$

Step 1: Analyze the function $g(x) = 2x^3 - 4x^2 - 6x$

• Degree: 3 (cubic), so the end behavior will be like $x^3$:

  • As $x \to -\infty$, $g(x) \to -\infty$
  • As $x \to \infty$, $g(x) \to \infty$

• Zeros: Factor the function:
  $g(x) = 2x(x^2 - 2x - 3)$
  Factor further:
  $g(x) = 2x(x - 3)(x + 1)$
  Zeros: $x = 0, 3, -1$

• Behavior at Zeros: Since these are all linear factors, the graph crosses the x-axis at each zero.

• Overall Shape: Typical cubic function with three x-intercepts and end behavior like $x^3$.

Step 2: Analyze the function $h(x) = -(x - 4)^3(x - 1)$

• Degree: $3 + 1 = 4$ (quartic), so the end behavior will be like $-x^4$:

  • As $x \to \pm\infty$, $h(x) \to -\infty$ (because of the negative sign)

• Zeros: $h(x) = 0$ when $x = 4$ (with multiplicity 3) and $x = 1$ (with multiplicity 1)

• Behavior at Zeros:

  • At $x = 4$, the graph touches the x-axis and turns around because of the cubic multiplicity.
  • At $x = 1$, the graph crosses the x-axis because of the linear factor.

• Overall Shape: The graph will have a turning point at $x = 4$, cross the x-axis at $x = 1$, and show downward behavior at both ends.

Step 3: Match Functions to Graphs

• Graph matching:
  • Graph B corresponds to $g(x) = 2x^3 - 4x^2 - 6x$ (cubic with three distinct x-intercepts and end behavior of a cubic).
  • Graph E corresponds to $h(x) = -(x - 4)^3(x - 1)$ (quartic with end behavior of $-x^4$, touches at $x = 4$, crosses at $x = 1$).

Final Answer:

• Graph B for $g(x)$
• Graph E for $h(x)$

Would you like further explanation on any step?

Here are 5 related questions you can explore:

1. How do you determine the end behavior of a polynomial function based on its degree?
2. Why does the multiplicity of a zero affect whether the graph crosses or touches the x-axis?
3. How can you factor a cubic polynomial to find its roots?
4. What is the significance of turning points in polynomial graphs?
5. How does a negative leading coefficient affect the overall shape of a polynomial graph?

Tip: Remember that polynomial graphs often reflect their degree and leading coefficient in their end behavior and general shape.