https://file.aiquickdraw.com/mathbot/file/1728497667060f8stls7g.jpg

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?

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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).

The graph shows the linear function f(x) = x and four transformations of f(x). Match each transformation to its corresponding graph.

This problem involves transformations of the linear function f(x) = x. Learn about horizontal and vertical shifts, as well as stretching and compression of the graph. Suitable for students in Grades 9-12.

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