In the provided images, the task is to identify the function family to which each function belongs and compare the graph of each function to the graph of its parent function.

1. First Function: $f(x) = 2|x + 2| - 8$

• Function Family: This function belongs to the absolute value function family.
• Parent Function: The parent function is $f(x) = |x|$.
• Comparison:
  • Translation: The graph is shifted 2 units to the left and 8 units down.
  • Stretching: The graph is vertically stretched by a factor of 2, making it narrower compared to the parent function.

2. Second Function: $f(x) = -2x^2 + 3$

• Function Family: This function belongs to the quadratic function family.
• Parent Function: The parent function is $f(x) = x^2$.
• Comparison:
  • Reflection: The graph is reflected over the x-axis because of the negative sign.
  • Stretching: The graph is vertically stretched by a factor of 2, making it narrower compared to the parent function.
  • Translation: The graph is shifted 3 units up.

Would you like further details or have any questions?

Related Questions:

1. How do vertical stretches and compressions affect the shape of a graph?
2. What are the general effects of horizontal translations on the graph of a function?
3. How do you determine if a function is reflected over the x-axis or y-axis?
4. What are the key characteristics of absolute value functions?
5. How do you determine the vertex of a quadratic function in standard form?

Tip: Always compare transformations to the parent function to better understand the graph's behavior.