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To find the \( f \)-intercept and \( t \)-intercepts for the given function \( f(t) = (t-2)(t+1)(t-4) \):

1. **\( f \)-intercept**: This is found by evaluating the function when \( t = 0 \).
\[ f(0) = (0-2)(0+1)(0-4) = (-2)(1)(-4) = 8 \]
So, the \( f \)-intercept is \( f(0) = 8 \).

2. **\( t \)-intercepts**: These are the values of \( t \) for which \( f(t) = 0 \). For the function \( f(t) = (t-2)(t+1)(t-4) \), setting each factor to zero gives:
   - \( t-2 = 0 \Rightarrow t = 2 \)
   - \( t+1 = 0 \Rightarrow t = -1 \)
   - \( t-4 = 0 \Rightarrow t = 4 \)
Thus, the \( t \)-intercepts are \( t = 2, -1, \) and \( 4 \).

Would you like more details on how these values are computed or have other questions about the function?

Here are some further questions to consider:
1. How would you find the maximum or minimum values of this function?
2. Can you graph the function based on the intercepts and factorization?
3. What does the end behavior of the function look like based on its degree and leading coefficient?
4. How would the function change if one of the factors was squared, such as \( (t-2)^2(t+1)(t-4) \)?
5. What are the possible symmetries in this polynomial function?

Tip: Remember, the \( f \)-intercept is always the value of the function at \( t = 0 \) while the \( t \)-intercepts are the values of \( t \) that make the function equal to zero.

Given the function f(t) = (t - 2)(t + 1)(t - 4), find its f-intercept and t-intercepts.

Learn how to determine the f-intercept and t-intercepts of the cubic polynomial f(t) = (t - 2)(t + 1)(t - 4). This guide provides a step-by-step solution using the Zero Product Property and basic polynomial factoring techniques, suitable for high school students.

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