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When graphing the function [tex]f(x) = -|x + 5| + 12[/tex] on your graphing calculator, what is the most appropriate viewing window for determining the domain and range of the function?

A. [tex]X_{\text{min}}: -10, X_{\text{max}}: 10[/tex]; [tex]Y_{\text{min}}: -10, Y_{\text{max}}: 10[/tex]
B. [tex]X_{\text{min}}: -20, X_{\text{max}}: 20[/tex]; [tex]Y_{\text{min}}: -20, Y_{\text{max}}: 20[/tex]
C. [tex]X_{\text{min}}: -5, X_{\text{max}}: 5[/tex]; [tex]Y_{\text{min}}: -20, Y_{\text{max}}: 20[/tex]
D. [tex]X_{\text{min}}: -10, X_{\text{max}}: 10[/tex]; [tex]Y_{\text{min}}: -5, Y_{\text{max}}: 5[/tex]

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GinnyAnswer
To determine the most appropriate viewing window for graphing the function [tex]\( f(x) = -|x+5| + 12 \)[/tex], let's analyze its domain and range.

1. Domain: The domain of [tex]\( f(x) = -|x+5| + 12 \)[/tex] includes all real numbers, because the absolute value function [tex]\( |x+5| \)[/tex] is defined for all [tex]\( x \)[/tex]. Therefore, [tex]\( x \)[/tex] can range from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].

2. Range:
- The vertex form of the absolute value function [tex]\( -|x+5| \)[/tex] reflects [tex]\( |x+5| \)[/tex] over the x-axis and shifts it horizontally by -5.
- The entire function [tex]\( f(x) = -|x+5| + 12 \)[/tex] shifts the function 12 units upward.
- The maximum value of [tex]\( -|x+5| \)[/tex] occurs when [tex]\( x = -5 \)[/tex], where [tex]\( -| -5 + 5 | = 0 \)[/tex].
- Therefore, [tex]\( f(-5) = -| -5 + 5 | + 12 = 0 + 12 = 12 \)[/tex].

Given the reflection and the shift, the maximum value is 12. For other values of [tex]\( x \)[/tex], [tex]\( -|x+5| \)[/tex] will be negative and decrease, shifting the values of [tex]\( f(x) \)[/tex] accordingly. Hence, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) \in (-\infty, 12] \][/tex]

Evaluating the viewing window:

1. Options for [tex]\( X_{\min} \)[/tex] and [tex]\( X_{\max} \)[/tex]:
- Given the domain is all real numbers, a typical interval of reasonable width should adequately cover it.
- An interval like [tex]\([-10, 10]\)[/tex] or [tex]\([-20, 20]\)[/tex] is sufficient because it includes typical graphing ranges and captures necessary details for visualization.
- Therefore, [tex]\(\boldsymbol{-10 \leq X \leq 10}\)[/tex] or [tex]\(\boldsymbol{-20 \leq X \leq 20}\)[/tex]

2. Options for [tex]\( Y_{\min} \)[/tex] and [tex]\( Y_{\max} \)[/tex]:
- The maximum value of [tex]\( f(x) = 12 \)[/tex].
- A [tex]\(\boldsymbol{Y_{\max}}\)[/tex] slightly above 12 is appropriate to avoid the graph being cut off at the top.
- Given the other available options, let’s choose [tex]\( \boldsymbol{(-20, 20})\)[/tex] to capture the essential behavior of the function and avoid cutting off any parts of the graph.

Having considered the domain and range, the most appropriate viewing window is:
[tex]\[ \boxed{X_{\min} = -10, \ X_{\max} = 10, \ Y_{\min} = -20, \ Y_{\max} = 20} \][/tex]
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