To evaluate the function $f(x) = -3(x - 4)^2(x + 2)$ at the indicated values, we’ll substitute $x = 0$ and $x = 3$ into the function and simplify.

Part 1: $f(0)$

Substitute $x = 0$ into the function: $f(0) = -3(0 - 4)^2(0 + 2)$ Simplify each part inside the parentheses: $= -3(-4)^2 \cdot 2$ $= -3 \cdot 16 \cdot 2$ $= -3 \cdot 32 = -96$ So, $f(0) = -96$.

Part 2: $f(3)$

Substitute $x = 3$ into the function: $f(3) = -3(3 - 4)^2(3 + 2)$ Simplify each part inside the parentheses: $= -3(-1)^2 \cdot 5$ $= -3 \cdot 1 \cdot 5$ $= -3 \cdot 5 = -15$ So, $f(3) = -15$.

Final Answers

• $f(0) = -96$
• $f(3) = -15$

Would you like further details on any part of this solution?

Here are some related questions to deepen understanding:

1. What happens to $f(x)$ when $x = 4$ or $x = -2$?
2. How would the function change if the coefficient $-3$ were positive?
3. How does the function behave as $x$ approaches very large positive or negative values?
4. What are the roots of $f(x)$ when set to zero?
5. How does squaring $(x - 4)$ affect the function's graph shape?

Tip: When evaluating functions, simplify each term step-by-step for clarity and accuracy.