https://file.aiquickdraw.com/mathbot/file/17292859603714hsr4djq.jpg

The graph of f (x)= x is transformed to create the graph of g
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x . Which graph best represents g?

The graph you provided shows a line that slopes downward from left to right, cutting through the origin. Based on this, let's consider the given transformations.

### Problem Interpretation:
You have the function \( f(x) = x \), which is the basic identity function with a slope of 1. The transformation given is \( g(x) = f\left(\frac{1}{2}x\right) \). This means that the graph has undergone a **horizontal stretch** by a factor of 2.

### Explanation of the Transformation:
- **Horizontal stretch**: When you apply \( f\left(\frac{1}{2}x\right) \), it stretches the graph horizontally by a factor of 2. This makes the graph appear wider, and the x-values need to increase more to get the same y-values. 
- In effect, the slope decreases (it visually becomes "flatter"), even though the y-values haven't changed.

### Comparing to the Graph:
Looking at the image of the graph, it appears to already represent this transformation. The line passes through points like \( (-6, 3) \) and \( (6, -3) \), and this flatter slope is consistent with the transformation \( g(x) = f\left(\frac{1}{2}x\right) \), which should have a slope of \( \frac{1}{2} \) instead of 1.

### Conclusion:
The graph shown in your image correctly represents the transformed function \( g(x) = f\left(\frac{1}{2}x\right) \). 

Would you like more details on how transformations affect the graph? 

Here are five related questions for further exploration:
1. What happens to the graph if the transformation is \( f(2x) \) instead of \( f\left(\frac{1}{2}x\right) \)?
2. How does a vertical stretch affect the slope of a linear graph?
3. What is the general form of a linear transformation for any function?
4. How would the graph of \( g(x) = -f\left(\frac{1}{2}x\right) \) look?
5. How do transformations differ between linear and non-linear functions?

**Tip:** When dealing with transformations, always focus on whether the change is applied inside the function (horizontal change) or outside the function (vertical change).

The graph of f(x) = x is transformed to create the graph of g(x) = f(1/2 x). Which graph best represents g?

Explore how the graph of the linear function f(x) = x transforms into g(x) = f(1/2 x), with a horizontal stretch by a factor of 2. Analyze the differences between the two graphs and learn the effects of this transformation. This guide is suitable for students in Grades 9-10 studying function transformations.

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