1. **Problem Statement:** Ronda plays a game guessing an integer between 0 and 100. Her score after one guess $x$ is given by the function $$f(x) = -|x - 57| + 300$$ where 57 is the randomly chosen integer. 2. **Function Explanation:** The function subtracts the distance between the guess $x$ and 57 from 300. The absolute value $|x - 57|$ measures how far the guess is from 57. 3. **Graphing the Function:** The graph is a "V" shape inverted (because of the negative sign) with vertex at $(57, 300)$. 4. **Key Features:** - **Vertex:** $(57, 300)$ is the maximum point. - **Domain:** $0 \leq x \leq 100$ (since guesses are between 0 and 100). - **Range:** $0 \leq f(x) \leq 300$ (score cannot be negative or exceed 300). - **x-intercepts:** Solve $f(x) = 0$: $$0 = -|x - 57| + 300 \implies |x - 57| = 300$$ Since $|x - 57|$ max is 57 (distance from 0 to 57 or 100 to 57 is less than 300), no real x-intercepts within domain. - **y-intercept:** At $x=0$: $$f(0) = -|0 - 57| + 300 = -57 + 300 = 243$$ 5. **Intervals of Increase and Decrease:** - Increasing on $[0, 57]$ because as $x$ approaches 57 from the left, $|x-57|$ decreases, so $f(x)$ increases. - Decreasing on $[57, 100]$ because as $x$ moves away from 57 to the right, $|x-57|$ increases, so $f(x)$ decreases. 6. **Intervals where $f(x)$ is positive or negative:** - $f(x)$ is positive for all $x$ in $[0, 100]$ because minimum value at endpoints is $f(0)=243$ and $f(100) = -|100-57| + 300 = -43 + 300 = 257$. - $f(x)$ is never negative in the domain. 7. **End Behavior:** - As $x \to 0$, $f(x) \to 243$. - As $x \to 100$, $f(x) \to 257$. 8. **Contextual Constraints:** - Guesses must be integers between 0 and 100. - Score cannot be negative. - Maximum score is 300 when guess is exactly 57. --- **Final answers:** - Vertex (maximum score): $(57, 300)$ - Increasing on $[0, 57]$ - Decreasing on $[57, 100]$ - No x-intercepts in domain - y-intercept at $(0, 243)$ - Score always positive in domain