The graph of the the function​ f(x) is shown at the right.
a. Make a table of values for​ f(x) and ​f(xplus5​).
b. Graph​ f(x) and ​f(xplus5​) on the same coordinate grid.

To solve this problem, we'll follow these steps:

### a. Make a Table of Values for \( f(x) \) and \( f(x+5) \)

1. **Choose values for \( x \)**: Typically, it's good to start by selecting a range of \( x \) values that covers the significant parts of the graph. Suppose you select values like \( -3, -2, -1, 0, 1, 2, 3 \).

2. **Calculate \( f(x) \) for each \( x \)**: Look at the graph to determine \( f(x) \) for each \( x \) value you selected.

3. **Calculate \( f(x+5) \)**: For each \( x \) value, substitute \( x+5 \) into the function and determine \( f(x+5) \) using the same method.

### Example Table:
Let's assume these sample values based on a generic graph. (You would need the exact graph to provide precise values.)

| \( x \) | \( f(x) \) | \( x+5 \) | \( f(x+5) \) |
|---------|------------|-----------|--------------|
| -3      | -1         | 2         | 3            |
| -2      | 0          | 3         | 2            |
| -1      | 1          | 4         | 1            |
| 0       | 2          | 5         | 0            |
| 1       | 3          | 6         | -1           |
| 2       | 2          | 7         | -2           |
| 3       | 1          | 8         | -3           |

### b. Graph \( f(x) \) and \( f(x+5) \) on the Same Coordinate Grid

To graph both \( f(x) \) and \( f(x+5) \):

1. **Plot the points for \( f(x) \)**: Use the original \( f(x) \) values.
2. **Plot the points for \( f(x+5) \)**: Use the transformed values.
3. **Draw the curves**: Connect the points smoothly for each function.

### Explanation:
- The function \( f(x+5) \) is a **horizontal shift** of \( f(x) \) by 5 units to the **left**.

If you want to upload the graph or provide specific values, I can give more precise results.

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Do you need more details or have any other questions?

Here are 5 related questions for further exploration:

1. What happens to the graph of \( f(x) \) if you consider \( f(x-5) \) instead of \( f(x+5) \)?
2. How does the graph of \( f(x) \) compare to the graph of \( -f(x) \)?
3. What is the effect of a vertical shift, such as \( f(x) + 3 \)?
4. How would you graph \( 2f(x) \) compared to \( f(x) \)?
5. Can you determine the equation of \( f(x) \) from its graph?

**Tip:** Understanding the effect of transformations on graphs (like shifts, stretches, and reflections) is crucial for mastering functions and their properties.

Explore how to create a table of values and graph functions f(x) and f(x+5) on the same coordinate grid. This problem involves analyzing the effect of horizontal shifts on function graphs, suitable for advanced high school students.

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