To solve this problem, let's analyze the graphs and understand the transformations applied to the function $f(x)$ to get $g(x)$.

Analyzing the Graphs:

1. First Graph: $f(x)$

   • The graph of $f(x)$ appears to be a "V" shape. It starts from point (0, 0), goes up to point (2, 4), and then goes back down to point (4, 0).

2. Second Graph: $g(x)$

   • The graph of $g(x)$ is also a "V" shape but looks different from $f(x)$. It starts from point (0, 0), goes down to point (1, -4), and then back up to point (2, 0).

Identifying the Transformation:

The problem states that $g(x)$ is a transformation of $f(x)$ in the form:

$g(x) = A f(Bx)$

Where $A$ and $B$ are constants.

Determining $A$ and $B$:

1. Vertical Stretch/Compression (A):

   • The maximum (and minimum) values of $f(x)$ are 4 and 0, respectively. For $g(x)$, the corresponding values are 0 and -4.
   • This change suggests a reflection over the x-axis and a vertical stretch by a factor of 4.
   • Therefore, $A = -2$.

2. Horizontal Stretch/Compression (B):

   • Comparing the x-values where the critical points occur:
     • In $f(x)$, the "V" shape has its vertex at (2, 0).
     • In $g(x)$, the "V" shape has its vertex at (1, 0).
     • This suggests a horizontal compression by a factor of 2. Thus, $B = 2$.

Hence, the transformation can be written as:

$g(x) = -2 f(2x)$

Conclusion:

The values for the transformation parameters are:

• $A = -2$
• $B = 2$

Would you like more details or have any questions about this solution?

Further Questions to Explore:

1. How do vertical and horizontal transformations affect the graph of a function?
2. What would happen if the values of $A$ and $B$ were different?
3. How can we determine reflections in function transformations?
4. Can you derive a general rule for identifying transformations from graphs?
5. What other types of transformations can be applied to a function?

Tip:

When analyzing function transformations, always look for changes in key points (like intercepts and extrema) to determine the nature of the transformation.