https://file.aiquickdraw.com/mathbot/file/1730483270150n3wsbdkt.jpg

https://file.aiquickdraw.com/mathbot/file/1730483270150xsbmlec2.jpg

To solve this problem, let's examine the images and equations provided. Here’s a step-by-step approach:

### Given Information
1. **Function \( f(x) \)**: The function \( f(x) \) in blue is defined as:
   \[
   f(x) = x + 2
   \]

2. **Function \( g(x) \)**: The function \( g(x) \) in red is given in the form:
   \[
   g(x) = (x + 2) + k
   \]
   Here, \( k \) is an unknown constant that we need to determine.

### Graph Interpretation
From the graph:
- The blue line representing \( f(x) = x + 2 \) intersects the \( y \)-axis at \( y = 2 \).
- The red line \( g(x) \) intersects the \( y \)-axis at a lower point than \( f(x) \). To determine \( k \), we can examine the vertical shift between the two lines.

### Solution
Since \( g(x) = (x + 2) + k \) represents a vertical shift of \( f(x) \), we can calculate \( k \) by measuring this vertical distance. Observing the graph:
- The \( y \)-intercept of \( f(x) = x + 2 \) is \( 2 \).
- The \( y \)-intercept of \( g(x) \) appears to be \( -4 \).

So:
\[
k = -4 - 2 = -6
\]

Thus, \( k = -6 \).

### Final Answer
The value of \( k \) is **-6**.

### Transformation Description
The function \( g(x) \) is a vertical translation of \( f(x) \) downward by 6 units. Therefore:
- \( g(x) = (x + 2) - 6 \) is obtained by shifting the graph of \( f(x) = x + 2 \) down by 6 units.

Would you like further details or have any questions?

### Related Questions
1. What effect does changing \( k \) have on the graph of \( g(x) \)?
2. How would the graph look if \( k \) were positive?
3. What if \( g(x) \) was shifted horizontally instead of vertically?
4. How would the slope of \( f(x) \) and \( g(x) \) affect their relative positions if they were changed?
5. Can we express the transformation in terms of a translation vector?

### Tip
When examining transformations, remember that changes outside the function (like \( +k \)) affect the \( y \)-direction, while changes inside the function (like \( x + h \)) affect the \( x \)-direction.

Find the value of k for the function g. Then describe the graph of g as a transformation of the graph of f.

Learn how to find the value of k for the function g(x) in relation to f(x), and understand the transformation of g(x) as a vertical shift of f(x). This problem involves concepts of linear functions and transformations, suitable for students in Grades 8-10.

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