The problem asks you to determine the transformation that takes $y = f(x)$ to $y = g(x)$ based on the given graph, and express $g(x)$ in the form $g(x) = a f(bx)$, where $a > 0$ and $b > 0$.

Analysis of the graph:

• The graph of $f(x)$ appears to be a V-shaped graph, which likely represents the absolute value function $f(x) = |x|$.
• The new graph for $g(x)$ looks like a compressed version of the original graph and has been scaled vertically and horizontally.

Key observations:

1. The vertex of both graphs remains at $(0, 0)$, meaning no horizontal or vertical shifts.
2. The slope of the new lines is steeper, indicating a vertical stretch or horizontal compression.

Horizontal Compression:

The graph is horizontally compressed because the new graph seems to reach the same values as the original graph in a shorter span. The original graph reaches 8 units at $x = 8$, but the new one reaches the same height at $x = 4$. This suggests a horizontal compression by a factor of 2.

So, $b = 2$ (since horizontal compression by a factor of 2 means you replace $x$ with $2x$).

Vertical Stretch:

The graph also seems to be stretched vertically. The new graph's slope is steeper than the original, indicating a vertical stretch. If the original graph's slope was 1, the new graph's slope appears to be 2, indicating a vertical stretch by a factor of 2.

Thus, $a = 2$.

Equation for $g(x)$:

The transformation can be represented as: $g(x) = 2 f(2x)$

This means the graph of $f(x)$ has undergone a horizontal compression by a factor of 2 and a vertical stretch by a factor of 2.

Let me know if you need further clarification or details! Here are five related questions you can explore:

1. What is the general form of a vertically stretched graph?
2. How does horizontal compression affect the period of periodic functions?
3. Can you identify the difference between horizontal and vertical transformations from an equation?
4. How do reflections across the x-axis and y-axis change the form of a function?
5. What happens when you combine multiple transformations (like reflection, stretch, and translation)?

Tip: Always start by identifying changes in key points, like the vertex or intercepts, to determine the type of transformation in a graph!