The graph shows four transformations of the linear function $f(x) = x$. These transformations are:

1. $f\left( \frac{1}{3}x \right)$ — Horizontal stretching
2. $2f(x)$ — Vertical stretching
3. $f(x - 7)$ — Horizontal shift to the right
4. $f(x) + 8$ — Vertical shift upwards

Let's analyze each transformation and its corresponding graph.

1. $f\left( \frac{1}{3}x \right)$

This transformation represents a horizontal stretch by a factor of 3. The graph becomes wider as the function changes slower with respect to $x$.

2. $2f(x)$

This is a vertical stretch by a factor of 2. The slope of the line doubles, making it steeper.

3. $f(x - 7)$

This transformation shifts the graph 7 units to the right. It is a horizontal shift of the line without altering its slope.

4. $f(x) + 8$

This transformation shifts the graph 8 units upwards, moving the entire line vertically.

Based on the graph, the following labels can be matched:

• $g(x)$ corresponds to $f(x - 7)$ since it shows the horizontal shift.
• $j(x)$ corresponds to $f\left( \frac{1}{3}x \right)$ due to the horizontal stretching.
• $k(x)$ corresponds to $f(x) + 8$, showing the vertical shift.
• $h(x)$ corresponds to $2f(x)$ due to the steeper slope (vertical stretch).

Would you like further details on these transformations or more in-depth analysis?

Here are 5 related questions to expand your understanding:

1. How would the graph change if the transformation was $f(3x)$?
2. What is the effect of negative coefficients on horizontal and vertical stretches?
3. How does the graph of $f(x - c) + d$ generally behave?
4. Can you describe how a vertical compression would look on the graph?
5. What are the differences between horizontal and vertical transformations in general?

Tip: Always remember that horizontal transformations affect the input (inside the function), and vertical transformations affect the output (outside the function).