Let's break down each function given in the image. I will describe the transformations applied to each parent function $f(x)$, as well as how the graph of $g(x)$ is transformed from its parent function.

24. $g(x) = 3|x| - 4$

• Parent function: $f(x) = |x|$ (Absolute value function)
• Transformations:
  1. Vertical stretch by a factor of 3 (the 3 in front of $|x|$).
  2. Vertical shift downward by 4 units (the $-4$).

25. $g(x) = 3\sqrt{x} + 8$

• Parent function: $f(x) = \sqrt{x}$ (Square root function)
• Transformations:
  1. Vertical stretch by a factor of 3 (the 3 in front of $\sqrt{x}$).
  2. Vertical shift upward by 8 units (the $+8$).

26. $g(x) = \frac{4}{x+1}$

• Parent function: $f(x) = \frac{1}{x}$ (Reciprocal function)
• Transformations:
  1. Horizontal shift left by 1 unit (the $+1$ inside the denominator).
  2. Vertical stretch by a factor of 4 (the 4 in the numerator).

27. $g(x) = 2|x - 6|$

• Parent function: $f(x) = |x|$
• Transformations:
  1. Horizontal shift right by 6 units (the $-6$ inside the absolute value).
  2. Vertical stretch by a factor of 2 (the 2 in front of $|x - 6|$).

28. $g(x) = -5|x - 2|$

• Parent function: $f(x) = |x|$
• Transformations:
  1. Horizontal shift right by 2 units (the $-2$ inside the absolute value).
  2. Vertical stretch by a factor of 5.
  3. Reflection across the x-axis (because of the negative sign in front of the 5).

29. $g(x) = -2|x + 5|$

• Parent function: $f(x) = |x|$
• Transformations:
  1. Horizontal shift left by 5 units (the $+5$ inside the absolute value).
  2. Vertical stretch by a factor of 2.
  3. Reflection across the x-axis (the negative sign in front of 2).

30. $g(x) = \frac{1}{6x} + 7$

• Parent function: $f(x) = \frac{1}{x}$
• Transformations:
  1. Horizontal compression by a factor of $1/6$ (the $6x$ in the denominator).
  2. Vertical shift upward by 7 units (the $+7$).

31. $g(x) = \frac{\sqrt{x + 3}}{4}$

• Parent function: $f(x) = \sqrt{x}$
• Transformations:
  1. Horizontal shift left by 3 units (the $+3$ inside the square root).
  2. Vertical compression by a factor of 4 (the division by 4).

Would you like further details on any of the transformations, or do you have any questions about these functions?

Here are 5 additional related questions to expand on this topic:

1. What is the parent function of $g(x) = -|x| + 3$, and how does it move?
2. How would the graph of $g(x) = 2\sqrt{x - 4} - 1$ be transformed?
3. What happens if the coefficient inside an absolute value function is negative, like in $g(x) = -|x - 2| + 5$?
4. How does horizontal stretching or compression affect the graph of a function?
5. What are the key differences between vertical and horizontal transformations?

Tip: Always remember that vertical transformations affect the output (y-values), while horizontal transformations affect the input (x-values).