To find the x-intercepts, set the quadratic equation equal to zero and solve for \( r \). The y-intercept is the value of the function when \( r = 0 \). The vertex can be found using the vertex formula for a quadratic equation, which is given by \( r = -b/(2a) \).
To find the x-intercepts, we solve the equation \( r^2 - 4r - 21 = 0 \). The solutions are \( r = -3 \) and \( r = 7 \). Thus, the x-intercepts are:
\[
\boxed{(-3, 0) \text{ and } (7, 0)}
\]
The y-intercept is found by evaluating the function at \( r = 0 \):
\[
y = 0^2 - 4(0) - 21 = -21
\]
Thus, the y-intercept is:
\[
\boxed{(0, -21)}
\]
The vertex of the quadratic function can be found using the formula \( r = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -4 \):
\[
r = -\frac{-4}{2 \cdot 1} = 2
\]
Substituting \( r = 2 \) back into the function to find the corresponding \( y \)-value:
\[
y = 2^2 - 4(2) - 21 = -25
\]
Thus, the vertex is:
\[
\boxed{(2, -25)}
\]
- x-intercepts: \(\boxed{(-3, 0) \text{ and } (7, 0)}\)
- y-intercept: \(\boxed{(0, -21)}\)
- vertex: \(\boxed{(2, -25)}\)
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