To determine the key features of the graph of the function [tex]\( f(x) = \sqrt{-x - 1} + 7 \)[/tex], we start by examining how it transforms the parent function [tex]\( f(x) = \sqrt{x} \)[/tex].
### 1. Horizontal Reflection
The term inside the square root is [tex]\(-x\)[/tex]. This indicates that there is a horizontal reflection. A horizontal reflection flips the graph across the y-axis. Therefore, the graph of [tex]\( \sqrt{x} \)[/tex] will be reflected horizontally.
### 2. Horizontal Shift
Next, we look at the term inside the square root. In [tex]\( \sqrt{-x - 1} \)[/tex], we see [tex]\(-x - 1\)[/tex]. This can be rewritten as [tex]\(-1(x + 1)\)[/tex], indicating a shift of the graph to the left by 1 unit. In other words, for the function to be zero inside the square root, [tex]\( -x - 1 = 0 \)[/tex] which simplifies to [tex]\( x = -1 \)[/tex]. Thus, the entire graph is shifted 1 unit to the left.
### 3. Vertical Shift
Finally, we observe the +7 outside the square root function. This indicates a vertical shift upward by 7 units. This means that every point on the graph of the function will be moved up by a distance of 7 units.
To summarize, the key features of the transformations on the parent function [tex]\( f(x) = \sqrt{x} \)[/tex] for the function [tex]\( f(x) = \sqrt{-x - 1} + 7 \)[/tex] are as follows:
1. Horizontal Reflection: The graph is reflected horizontally across the y-axis.
2. Horizontal Shift: The graph is shifted 1 unit to the left.
3. Vertical Shift: The graph is shifted 7 units upward.
These transformations together describe the new graph of the function [tex]\( f(x) = \sqrt{-x - 1} + 7 \)[/tex].
