https://file.aiquickdraw.com/mathbot/file/1728707829975vvw77h5m.jpg

The problem given is:

The function is \( f(r) = (r - 3)(r + 1)(r - 7) \).

You are asked to find:
1. The coordinates of the \( f \)-intercept.
2. The coordinates of the \( r \)-intercepts.

### Solution:
1. **\( f \)-intercept** (or y-intercept):
   To find the \( f \)-intercept, set \( r = 0 \) and solve for \( f(r) \).

   \[
   f(0) = (0 - 3)(0 + 1)(0 - 7) = (-3)(1)(-7) = 21
   \]

   So, the \( f \)-intercept is \( (0, 21) \).

2. **\( r \)-intercepts** (where the function crosses the \( r \)-axis):
   To find the \( r \)-intercepts, set \( f(r) = 0 \) and solve for \( r \).

   The function is factored, so the \( r \)-intercepts occur where each factor equals zero:
   \[
   r - 3 = 0 \quad \Rightarrow \quad r = 3
   \]
   \[
   r + 1 = 0 \quad \Rightarrow \quad r = -1
   \]
   \[
   r - 7 = 0 \quad \Rightarrow \quad r = 7
   \]

   So, the \( r \)-intercepts are \( (3, 0) \), \( (-1, 0) \), and \( (7, 0) \).

### Final Answer:
- \( f \)-intercept: \( (0, 21) \)
- \( r \)-intercepts: \( (3, 0), (-1, 0), (7, 0) \)

Would you like further clarification or more details on any step?

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Here are some related questions you might want to explore:
1. How do you find the x-intercepts of a quadratic function?
2. What is the significance of the y-intercept in a polynomial function?
3. How can you factor a cubic polynomial if it’s not already factored?
4. What is the general shape of the graph of a cubic polynomial?
5. How can you verify the intercepts of a function graphically?

**Tip:** When looking for intercepts, always set the opposite variable to zero (e.g., for the x-intercept, set y = 0).

Given the function f(r) = (r - 3)(r + 1)(r - 7), find the f-intercept and the r-intercepts.

Learn how to find the f-intercept and r-intercepts of the polynomial function f(r) = (r - 3)(r + 1)(r - 7). This problem demonstrates factoring and solving for intercepts, suitable for Grades 10-12 students.

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