Let's analyze the two given polynomial functions to determine their corresponding graphs:

1. Function $g(x) = x^4 - 9x^2$:

   • This is a quartic function (degree 4), and it can be factored as: $g(x) = x^2(x^2 - 9) = x^2(x - 3)(x + 3)$
     • The function has roots at $x = -3, 0, 3$.
     • As it's a positive quartic function, it will have an "M" or "W" shape.
     • The leading coefficient of $x^4$ is positive, so the graph will open upwards.

2. Function $h(x) = -2x(x^2 - 9) = -2x(x - 3)(x + 3)$:

   • This is a cubic function (degree 3).
   • It has roots at $x = -3, 0, 3$, similar to $g(x)$, but since the leading coefficient is negative, the graph will have a shape where it falls from the left and rises to the right (an "S"-shaped curve).
   • The negative coefficient causes the graph to be flipped.

Matching graphs to functions:

• Graph A corresponds to $g(x) = x^4 - 9x^2$ because it has the "W" shape and intersects the x-axis at $x = -3, 0, 3$.
• Graph E corresponds to $h(x) = -2x(x^2 - 9)$ because it shows the typical cubic behavior with roots at $x = -3, 0, 3$, and the graph decreases on the left side and increases on the right side.

Final Answers:

• (a) The graph of $g(x) = x^4 - 9x^2$ is Graph A.
• (b) The graph of $h(x) = -2x(x^2 - 9)$ is Graph E.

Would you like further details or explanations? Here are five follow-up questions to expand understanding:

1. How do you determine the degree of a polynomial from its equation?
2. Why does a positive leading coefficient in a quartic function cause the graph to open upward?
3. What is the significance of factoring a polynomial in graph analysis?
4. How do transformations affect the shape and direction of polynomial graphs?
5. What are the general characteristics of cubic vs. quartic functions in terms of graph shapes?

Tip: Always look at the degree and leading coefficient of a polynomial to predict its end behavior!