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?

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).