Let's analyze each of the functions and determine their parent functions along with how they have been transformed.

Problem 24: $g(x) = 3|x| - 4$

• Parent Function: $f(x) = |x|$
• Transformations:
  • Vertical stretch by a factor of 3.
  • Shifted 4 units downward.

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

• Parent Function: $f(x) = \sqrt{x}$
• Transformations:
  • Vertical stretch by a factor of 3.
  • Shifted 8 units upward.

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

• Parent Function: $f(x) = \frac{1}{x}$
• Transformations:
  • Vertical stretch by a factor of 4.
  • Shifted 1 unit to the left.

Problem 27: $g(x) = 2|x-6|$

• Parent Function: $f(x) = |x|$
• Transformations:
  • Vertical stretch by a factor of 2.
  • Shifted 6 units to the right.

Problem 28: $g(x) = -5[x - 2]$

• Parent Function: $f(x) = x$ (assuming this is a linear function).
• Transformations:
  • Vertical stretch by a factor of 5.
  • Reflected across the x-axis (due to the negative sign).
  • Shifted 2 units to the right.

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

• Parent Function: $f(x) = |x|$
• Transformations:
  • Vertical stretch by a factor of 2.
  • Reflected across the x-axis (due to the negative sign).
  • Shifted 5 units to the left.

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

• Parent Function: $f(x) = \frac{1}{x}$
• Transformations:
  • Horizontal stretch by a factor of 6 (since $6x$ compresses horizontally).
  • Shifted 7 units upward.

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

• Parent Function: $f(x) = \sqrt{x}$
• Transformations:
  • Shifted 3 units to the left.
  • Vertical shrink by a factor of 4.

Would you like more details on any of these transformations?

Related questions:

1. How does a vertical stretch affect the graph of a function?
2. What does it mean when a function is reflected across the x-axis?
3. How do horizontal shifts differ from vertical shifts in terms of transformations?
4. Why does multiplying by a constant affect the slope or curvature of functions differently for linear and nonlinear parent functions?
5. How can we determine if a shift occurs to the left or right by analyzing the equation?

Tip:

Always check inside the function's parentheses for horizontal shifts and outside for vertical shifts.