Let's work through each problem step by step from the image you uploaded.

1. Problem Breakdown:

21. $g(x) = 12 - x^2$

• a. Identify the parent function: The parent function here is $f(x) = x^2$ (a basic quadratic function).

• b. Describe the sequence of transformations from $f$ to $g$:

  1. Reflection across the x-axis because of the negative sign in front of $x^2$.
  2. Vertical translation upwards by 12 units due to the "+12".

• c. Sketch the graph of $g$: The graph is an upside-down parabola (due to the reflection) shifted 12 units upwards.

• d. Use function notation to write $g$ in terms of $f$: If $f(x) = x^2$, then $g(x) = 12 - f(x)$.

23. $g(x) = x^3 + 7$

• a. Identify the parent function: The parent function is $f(x) = x^3$ (a cubic function).

• b. Describe the sequence of transformations:

  1. Vertical translation upwards by 7 units.

• c. Sketch the graph of $g$: The graph looks like a cubic curve shifted 7 units upwards.

• d. Use function notation to write $g$ in terms of $f$: If $f(x) = x^3$, then $g(x) = f(x) + 7$.

27. $g(x) = 2 - (x + 5)^2$

• a. Identify the parent function: The parent function is $f(x) = x^2$ (quadratic).

• b. Describe the sequence of transformations:

  1. Horizontal shift left by 5 units because of $(x + 5)$.
  2. Reflection across the x-axis because of the negative in front of $(x + 5)^2$.
  3. Vertical translation upwards by 2 units.

• c. Sketch the graph of $g$: An upside-down parabola (due to the reflection) shifted 5 units to the left and 2 units upwards.

• d. Use function notation to write $g$ in terms of $f$: If $f(x) = x^2$, then $g(x) = 2 - f(x+5)$.

31. $g(x) = (x - 1)^3 + 2$

• a. Identify the parent function: The parent function is $f(x) = x^3$ (cubic).

• b. Describe the sequence of transformations:

  1. Horizontal shift right by 1 unit because of $(x - 1)$.
  2. Vertical translation upwards by 2 units.

• c. Sketch the graph of $g$: A cubic curve shifted 1 unit to the right and 2 units upwards.

• d. Use function notation to write $g$ in terms of $f$: If $f(x) = x^3$, then $g(x) = f(x - 1) + 2$.

Do you want more details or have any questions?

Here are 5 related questions:

1. What are the effects of horizontal and vertical shifts on a function’s graph?
2. How does a reflection across the x-axis affect the graph of a function?
3. Can you describe the transformations for the function $g(x) = -x^2 - 3x + 5$?
4. What happens when a cubic function is reflected across both the x-axis and y-axis?
5. How does a vertical stretch or compression affect a quadratic function?

Tip: When analyzing transformations, always handle reflections and shifts carefully to avoid confusion.