The function you've provided is \( f(c) = y(y - 3)^2 \), where \( f \) is a function of \( c \), but it seems to be expressed in terms of \( y \). To provide a deeper analysis, it’s important to clarify whether there is any explicit relationship between \( y \) and \( c \), or if \( f(c) \) is indirectly defined via \( y \). However, assuming you are asking for a breakdown of \( f(y) = y(y - 3)^2 \), let's proceed with that.

### Simplifying \( f(y) = y(y - 3)^2 \)

1. Start by expanding \( (y - 3)^2 \):
   \[
   (y - 3)^2 = y^2 - 6y + 9
   \]

2. Now, substitute back into the function:
   \[
   f(y) = y(y^2 - 6y + 9)
   \]

3. Distribute \( y \) across the terms inside the parentheses:
   \[
   f(y) = y^3 - 6y^2 + 9y
   \]

Thus, the simplified form of the function is:
\[
f(y) = y^3 - 6y^2 + 9y
\]

### Next steps
This expression can be further analyzed in different contexts, such as finding its roots, graphing, or determining maxima and minima. Let me know how you'd like to proceed!

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Would you like to:
1. Find the critical points (derivatives)?
2. Solve \( f(y) = 0 \)?
3. Explore the graph of the function?
4. Investigate the behavior of the function for large values of \( y \)?
5. Discuss any particular applications of this function?

**Tip:** When working with polynomial functions, understanding their degree helps predict the number of possible turning points (up to \( n-1 \) for a degree \( n \) polynomial).

Understand how to simplify the polynomial function f(c) = y(y-3)². This problem covers algebraic concepts such as expanding and simplifying polynomials. Follow step-by-step solutions to express the function in its simplest form. Ideal for students in Grades 10-12.

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