Here is a graph of a segment of
\[f(x)=x\].
\[g(x)=f(x+7)\]
Graph
\[g\].

To graph the function \(g(x) = f(x+7)\), we need to understand how the transformation \(f(x + 7)\) affects the graph of \(f(x) = x\).

### Step-by-Step Explanation:
1. **Original Function \(f(x) = x\):**
   - The graph of \(f(x) = x\) is a straight line passing through the origin with a slope of 1.
   - Its equation tells us that for each point \((x, f(x))\), the corresponding output is the same as the input, so the line rises at a 45-degree angle.

2. **Transformation \(g(x) = f(x+7)\):**
   - This represents a horizontal shift.
   - The graph of \(f(x + 7)\) shifts the original graph of \(f(x) = x\) to the **left by 7 units**.
   - Every point on the graph of \(f(x)\) is now shifted 7 units leftward.

### Graphing:
- The line \(f(x) = x\) has points like \((0, 0)\), \((1, 1)\), and \((2, 2)\).
- After the shift, the new graph \(g(x) = f(x + 7)\) will have points such as:
  - \((-7, 0)\), \((-6, 1)\), and \((-5, 2)\).
- The slope remains the same (1), but the entire graph is moved to the left by 7 units.

Would you like me to create a visual representation of this graph?

### Additional Questions:
1. How would \(g(x) = f(x - 7)\) shift the graph?
2. What happens if we add a vertical shift, say \(g(x) = f(x+7) + 3\)?
3. How would the transformation affect the slope of the function if we had \(f(x) = 2x\)?
4. How do we represent a vertical reflection in such a graph?
5. How does a combination of horizontal and vertical shifts change the graph?

### Tip:
Horizontal shifts occur when the transformation is applied inside the function (with \(x\)), while vertical shifts occur outside the function.

Here is a graph of a segment of f(x)=x. g(x)=f(x+7) Graph g.

Understand how to graph the transformation g(x) = f(x+7) by shifting the linear function f(x) = x horizontally. This problem explains the concept of horizontal shifts and their effect on the graph of linear equations, suitable for students in Grades 9-10.

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