Let's work through these problems step-by-step.

Problem 11:

Parent function appears to be $f(x) = |x|$ (absolute value function).

• a) Parent Function: $f(x) = |x|$
• b) Translate 4 units right: The function becomes $f(x) = |x - 4|$
• c) Reflect in the x-axis and translate 5 units up: Reflecting in the x-axis changes the function to $f(x) = -|x|$, then translating 5 units up gives $f(x) = -|x| + 5$

Problem 12:

Parent function appears to be $f(x) = \frac{1}{x}$ (rational function).

• a) Parent Function: $f(x) = \frac{1}{x}$
• b) Translate 7 units down and 3 units left: The function becomes $f(x) = \frac{1}{x + 3} - 7$
• c) Horizontally stretch by a factor of 3 and translate 2 units right: Horizontally stretching changes the function to $f(x) = \frac{1}{\frac{x}{3}} = \frac{3}{x}$, and then translating 2 units right gives $f(x) = \frac{3}{x - 2}$

Problem 13:

Parent function appears to be $f(x) = x$ (linear function).

• a) Parent Function: $f(x) = x$
• b) Translate 1 unit up and vertically stretch by a factor of 4: First stretch vertically to get $f(x) = 4x$, then translate 1 unit up: $f(x) = 4x + 1$
• c) Vertically compress by a factor of 1/3 and translate 6 units left: Vertically compress to get $f(x) = \frac{1}{3}x$, then translate 6 units left: $f(x) = \frac{1}{3}(x + 6)$

Problem 14:

Parent function appears to be $f(x) = \sqrt{x}$ (square root function).

• a) Parent Function: $f(x) = \sqrt{x}$
• b) Reflect in the y-axis and translate 4 units right: Reflecting in the y-axis changes the function to $f(x) = \sqrt{-x}$, then translating 4 units right gives $f(x) = \sqrt{-(x - 4)}$
• c) Horizontally compress by a factor of 1/2 and translate 5 units down: Horizontally compress to get $f(x) = \sqrt{2x}$, then translate 5 units down: $f(x) = \sqrt{2x} - 5$

Problem 15:

You are given $f(x) = 2(x - 5)^2 + 3$. First, reflect across the y-axis, which gives $f(x) = 2(-x - 5)^2 + 3$. Then, translate 6 units right and 2 units down, resulting in: $f(x) = 2(-(x - 6) - 5)^2 + 3 - 2$ Simplifying: $f(x) = 2(-x + 1)^2 + 1$

Problem 16:

The resulting function is $f(x) = 6|x + 4| - 1$. To find the original function, reverse the transformations:

1. Reverse the vertical stretch by a factor of 2: $\frac{1}{2} \times 6 = 3$, so $f(x) = 3|x + 4| - 1$
2. Reverse the translation of 5 units up: $f(x) = 3|x + 4| - 5$
3. Reverse the translation of 2 units left: $f(x) = 3|x + 2| - 5$

The original function is $f(x) = 3|x + 2| - 5$.

Do you want further details on any of these transformations?
Here are five related questions to practice:

1. How do you translate a function vertically by 7 units?
2. What is the impact of a reflection across the x-axis on a function?
3. What does it mean to horizontally stretch a function by a factor of 2?
4. How would you reflect a function over the y-axis?
5. What are the effects of compressing a function vertically?

Tip: When translating a function, make sure to adjust the function's input for horizontal changes and the output for vertical changes.