To determine the most appropriate viewing window for the function \( f(x) = -|x + 5| + 12 \), we need to focus on the range of \( x \) values that encompass the significant features of the function and an appropriate range for \( y \) values to capture the maximum and minimum values of the function.
First, let's analyze the function \( f(x) = -|x + 5| + 12 \).
1. Domain Considerations:
- The domain of \( f(x) \) is all real numbers because the absolute value function \( |x+5| \) and its transformations do not place any restrictions on \( x \). Thus, \( x \) can range from negative to positive infinity.
- However, for practical purposes of plotting and visualizing the graph, we generally choose a finite interval that captures the most interesting parts of the graph.
2. Range Considerations:
- The function \( f(x) = -|x + 5| + 12 \) is a V-shaped graph that opens downward, with its vertex at \( x = -5 \). The vertex form of this function helps show that the maximum value of \( f(x) \) is \( 12 \), occurring when \( x = -5 \).
- As \( x \) moves away from \( -5 \) in either direction, \( f(x) \) decreases.
Now, let's look at the suggested window options:
- \(X \min: -10, X \max: 10; Y \min: -10, Y \max: 10\)
- \(X \min: -20, X \max: 20; Y \min: -20, Y \max: 20\)
- \( X \min: -5, X \max: 5; Y \min: -20, Y \max: 20 \)
- \( X \min: -10, X \max: 10; Y \min: -5, Y \max: 5 \)
Given that \( f(x) \) has a maximum value of \( 12 \) at \( x = -5 \), and because the function is symmetric around its vertex at \( x = -5 \), a horizontal range for \( x \) from \(-10\) to \(10\) captures the significant changes in the function value. Additionally, the vertical range for \( y \) from \(-10\) to \(10\) will adequately display the maximum value of \(12\) and cover enough range to observe the function's decrease.
Thus, the most appropriate viewing window is:
[tex]\[X \min = -10, X \max = 10, Y \min = -10, Y \max = 10\][/tex]
This viewing window ensures that both the domain and range are well-covered.
Therefore, the correct option is:
[tex]\[ \boxed{X\min: -10, X\max: 10, Y\min: -10, Y\max: 10} \][/tex]
