To solve this problem, we need to determine the x-intercepts and y-intercepts of the given function:

$r(x) = -x^2 + 45$

Steps:

1. Finding the x-intercept(s):

   • The x-intercept occurs when $r(x) = 0$, i.e., the output is 0.
   • Set the equation to 0: $-x^2 + 45 = 0$ Rearrange: $x^2 = 45$ Solve for $x$: $x = \pm \sqrt{45}$ Simplify: $x = \pm 3\sqrt{5}$ Thus, the x-intercepts are: $(3\sqrt{5}, 0) \text{ and } (-3\sqrt{5}, 0)$

2. Finding the y-intercept(s):

   • The y-intercept occurs when $x = 0$, i.e., the input is 0.
   • Substitute $x = 0$ into the function: $r(0) = -0^2 + 45 = 45$ Thus, the y-intercept is: $(0, 45)$

Final Answers:

• x-intercepts: $(3\sqrt{5}, 0)$ and $(-3\sqrt{5}, 0)$
• y-intercept: $(0, 45)$

Would you like further details or explanations? Let me know!

Relative Questions:

1. What does the square root symbol ($\sqrt{}$) represent, and how is it calculated?
2. How do you identify whether a quadratic equation has two, one, or no x-intercepts?
3. Why does the graph of $r(x) = -x^2 + 45$ open downward?
4. How does the value of 45 as a constant affect the graph of this function?
5. What is the relationship between x-intercepts and the roots of a quadratic equation?

Tip:

For any quadratic function $y = ax^2 + bx + c$, the y-intercept is always the value of $c$, and the x-intercepts can be calculated by solving the equation $ax^2 + bx + c = 0$.