https://file.aiquickdraw.com/mathbot/file/1724380605880v2vv99yr.jpg

The functions f and g are graphed in the same rectangular coordinate​ system, shown to the right. If g is obtained from f through a sequence of​ transformations, find an equation for g.What is the equation for​ g?

To determine the equation of \( g(x) \) from \( f(x) \), let's analyze the transformations applied to the graph of \( f(x) \) to obtain the graph of \( g(x) \).

Given the image, \( f(x) \) is a square root function, likely \( f(x) = \sqrt{x} \). The graph of \( g(x) \) is shifted upwards and possibly stretched.

### Step-by-Step Analysis:

1. **Identify the basic function \( f(x) \):**
   \[
   f(x) = \sqrt{x}
   \]
   This corresponds to the black graph in the image.

2. **Vertical shift:**
   The blue graph \( g(x) \) appears to be a vertical translation of \( f(x) \). It has been shifted up by 4 units.

   Therefore, the equation for \( g(x) \) based on this shift is:
   \[
   g(x) = \sqrt{x} + 4
   \]

Thus, the equation for \( g(x) \) is:
\[
g(x) = \sqrt{x} + 4
\]

Would you like more details on any specific part of this transformation process or have any further questions?

### Related Questions:
1. What would be the equation if \( g(x) \) was shifted 3 units down instead of up?
2. How does a horizontal shift affect the equation of a square root function?
3. What transformation occurs if \( g(x) = 2\sqrt{x} \)?
4. How do combined transformations (e.g., vertical shift and horizontal stretch) affect the function?
5. If \( g(x) = \sqrt{x - 2} + 4 \), what transformation does \( -2 \) represent?

### Tip:
When identifying transformations, always consider how each parameter (vertical shift, horizontal shift, stretching, reflection) affects the function's graph.

Learn how to find the equation of g(x) transformed from f(x) using vertical shift and possibly stretching. Detailed step-by-step analysis of functions in a rectangular coordinate system.

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