The given function $g(x) = -2|x - 4| + 3$ can be described as a transformation of the basic absolute value function $f(x) = |x|$. To understand the transformations applied, let's break down the expression step by step:

1. Horizontal Shift: The expression $|x - 4|$ shifts the graph horizontally by 4 units to the right. This is because of the $x - 4$ inside the absolute value.

2. Vertical Stretch and Reflection: The coefficient $-2$ outside the absolute value causes two effects:

   • The factor of $2$ stretches the graph vertically by a factor of 2.
   • The negative sign causes a reflection over the x-axis.

3. Vertical Shift: The $+3$ outside the absolute value shifts the graph vertically by 3 units upward.

Answer:

The transformations involved in obtaining $g(x)$ from $f(x)$ are:

• A. Reflection about the x-axis (due to the negative sign)
• B. Horizontal shift (4 units to the right)
• C. Vertical shift (3 units up)
• D. Vertical stretch/shrink (stretch by a factor of 2)

Would you like more details on any of these transformations or explanations?

Follow-up Questions:

1. How does changing the coefficient of the absolute value function affect its graph?
2. What is the effect of reflecting a graph over the x-axis versus the y-axis?
3. Can you explain how the horizontal and vertical shifts are determined in absolute value functions?
4. How would the graph of $g(x) = -|x - 4| + 3$ differ from the current function?
5. What happens if we change the vertical stretch factor from 2 to another value?

Tip: Always remember that terms inside the absolute value affect the graph horizontally, while terms outside affect it vertically.