Let's break down the functions you've given:

1. Function $f(x)$

The function $f(x) = x^2 - 6x + 9$ is a quadratic function.

We can rewrite it in factored form: $f(x) = (x - 3)^2$ This is a perfect square trinomial, so the vertex of this quadratic function is at $x = 3$, and the minimum value is at $f(3) = 0$.

2. Function $R(x)$

The function $R(x) = x^2(x)$ seems to indicate multiplication of $x^2$ and $x$, which gives: $R(x) = x^3$ So, $R(x)$ is a cubic function.

To summarize:

• $f(x) = (x - 3)^2$ is a parabola with a vertex at $x = 3$.
• $R(x) = x^3$ is a cubic function.

Would you like further details on how to graph these functions, compare their behaviors, or analyze their intersections?

Follow-up questions:

1. What is the vertex and axis of symmetry of $f(x)$?
2. What are the critical points of $R(x) = x^3$?
3. Would you like help finding the intersection points of $f(x)$ and $R(x)$?
4. How does the growth rate of $f(x)$ compare to $R(x)$?
5. Would you like to explore transformations of either function?

Tip: When analyzing the shape of functions, it's useful to start with identifying critical points such as the vertex for quadratics and inflection points for cubics.