Question 14 (1 point) Given the function $$ f(x)=\sqrt{x-h}+k $$ with a domain of $$ \{x \mid x \geq-5, x \in R\} $$ and a range of $$ \{y \mid y \geq 8, y \in R\} $$, which of the following best describes the vertical and horizontal translations with respect to the graph of $$ f(x)=\sqrt{x} $$ ? 5 units to left and 8 units up 8 units to left and 5 units down to left and 5 units up 5 units to left and 8 units down Question 15 (1 point)

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To analyze the function $$ f(x) = \sqrt{x - h} + k $$ and its translations with respect to the base function $$ f(x) = \sqrt{x} $$, we need to determine the values of $$ h $$ and $$ k $$ based on the given domain and range.

1. **Domain**: The function has a domain of $$ \{x \mid x \geq -5\} $$. This means that the expression inside the square root must be non-negative:
   $$
   x - h \geq 0 \implies x \geq h
   $$
   For the domain to start at $$ -5 $$, we set $$ h = -5 $$.

2. **Range**: The function has a range of $$ \{y \mid y \geq 8\} $$. The minimum value of $$ f(x) $$ occurs when $$ x = h $$:
   $$
   f(h) = \sqrt{h - h} + k = 0 + k = k
   $$
   For the range to start at $$ 8 $$, we set $$ k = 8 $$.

Now we have:
- $$ h = -5 $$
- $$ k = 8 $$

The function can be rewritten as:
$$
f(x) = \sqrt{x + 5} + 8
$$

### Translations:
- The term $$ \sqrt{x + 5} $$ indicates a horizontal translation of **5 units to the left** (since $$ h = -5 $$).
- The term $$ +8 $$ indicates a vertical translation of **8 units up** (since $$ k = 8 $$).

### Conclusion:
The correct description of the translations with respect to the graph of $$ f(x) = \sqrt{x} $$ is:
**5 units to the left and 8 units up.**
5 units to the left and 8 units up

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